......!!!!!!!!...................................
09:18:13 query R.1.14 (was R.1.6) Of the numbers in the set {-sqrt(2), pi + sqrt(2), 1 / 2 + 10.3} which are counting numbers, which are rational numbers, which are irrational numbers and which are real numbers?
......!!!!!!!!...................................
RESPONSE --> None of these are counting numbers. Rational numbers are 1/2 and 10.3. Irrational numbers are -sqrt(2), pi+sqrt(2). All the numbers listed are real numbers.
.................................................
......!!!!!!!!...................................
09:18:32 ** Counting numbers are the numbers 1, 2, 3, .... . None of the given numbers are counting numbers Rational numbers are numbers which can be expressed as the ratio of two integers. 1/2+10.3 are rational numbers. Irrational numbers are numbers which can be expressed as the ratio of two integers. {-sqrt(2)}, pi+sqrt(2) are irrational numbers.. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:19:20 query R.1.32 (was R.1.24) Write in symbols: The product of 2 and x is the product of 4 and 6
......!!!!!!!!...................................
RESPONSE --> 2x=4(6)
.................................................
......!!!!!!!!...................................
09:19:41 ** The product of 2 and x is 2 * x and the product of 4 and 6 iw 4 * 6. To say that these are identical is to say that 2*x=4*6. **
......!!!!!!!!...................................
RESPONSE --> Thank you- I was not sure what symbol to use.
.................................................
......!!!!!!!!...................................
09:21:57 query R.1.50 (was R.1.42) Explain how you evaluate the expression 2 - 5 * 4 - [ 6 * ( 3 - 4) ]
......!!!!!!!!...................................
RESPONSE --> Innermost Parentheses are evaluated first= 3-4= -1. Then the next set of parentheses- 6 * -1= -6. Then Multiply 5*4= 20. Work the expression now from left to right= 2-20- (-6). 2-20= -18 - (-6)= -18+6= -12
.................................................
......!!!!!!!!...................................
09:22:12 **Starting with 2-5*4-[6*(3-4)]. First you evaluate the innermost group to get 2-5*4-[6*-1] . Then multiply inside brackets to get 2-5*4+6. Then do the multiplication to get 2-20+6. Then add and subtract in order, obtaining -12. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:24:58 query R.1.80 (was R.1.72) Explain how you use the distributive property to remove the parentheses from the express (x-2)(x-4).
......!!!!!!!!...................................
RESPONSE --> Break apart each item in the first set of parentheses and multiply it by each item in the second set of parentheses. x(x-4)= x squared --4x -2(x-4)= -2x+8 Add the two answers together: x squared-6x+8
.................................................
......!!!!!!!!...................................
09:25:44 ** COMMON ERROR: Using FOIL. FOIL is not the Distributive Law. FOIL works for binomial expressions. FOIL follows from the distributive law but is of extremely limited usefulness and the instructor does not recommend relying on FOIL. Starting with (x-2)(x-4) ; one application of the Distributive Property gives you x(x-4) - 2(x-4) . Applying the property to both of the other terms we get x^2 - 4x - (2x -8). Simplifying: x^2 - 4x - 2x + 8 or x^2 - 6x + 8. *
......!!!!!!!!...................................
RESPONSE --> okay- again I needed to use a different symbol for "squared", but overall, I got the same thing.
.................................................
......!!!!!!!!...................................
09:27:20 query R.1.86 (was R.1.78) Explain why (4+3) / (2+5) is not equal to 4/2 + 3/5.
......!!!!!!!!...................................
RESPONSE --> 4+3=7/ 2+5=7= 7/7=1 4/2+3/5= 20/10+6/10= 26/10 1 does not equal 26/10
.................................................
......!!!!!!!!...................................
09:27:34 ** Good answer but at an even more fundamental level it comes down to order of operations. (4+3)/(2+5) means 7/7 which is equal to 1. By order of operations, in which multiplications and divisions precede additions and subtractions, 4/2+3/5 means (4/2) + (3/5), which gives us 2+3/5 = 2 3/5 **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:28:11 Query Add comments on any surprises or insights you experienced as a result of this assignment.
......!!!!!!!!...................................
RESPONSE --> I learned the correct symbol for squared ^ and multiply *
.................................................
IYHeЁxl̻˨ assignment #002 C{ꃌtKޡY College Algebra 08-30-2005
......!!!!!!!!...................................
09:29:24 query R.2.46 (was R.2.36) Evaluate for x = -2, and y = 3 the expression (2x - 3) / y and explan how you got your result.
......!!!!!!!!...................................
RESPONSE --> (2(-2)-3)/3= -4-3/3= -7/3
.................................................
......!!!!!!!!...................................
09:29:31 ** Starting with (2x-3)/y we substitute x=-2 and y=3 to get (2*(-2) - 3)/3 = (-4-3)/3= -7/3. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:31:20 query R.2. 55 (was R.2.45) Evaluate for x = 3 and y = -2: | |4x| - |5y| | and explan how you got your result.
......!!!!!!!!...................................
RESPONSE --> I I4(3)I- I5(-2)I= I12I- I-10I = 12-10 (absolute value of a negative number is positive)= 2
.................................................
......!!!!!!!!...................................
09:31:29 ** Starting with | | 4x |- | 5y | | we substitute x=3 and y=-2 to get | | 4*3 | - | 5*-2 | | = | | 12 | - | -10 | | = | 12-10 | = | 2 | = 2. **
......!!!!!!!!...................................
RESPONSE --> okay
.................................................
......!!!!!!!!...................................
09:32:15 query R.2.64 (was R.2.54) Explain what values, if any, must not be present in the domain of the expression (-9x^2 - x + 1) / (x^3 + x)
......!!!!!!!!...................................
RESPONSE --> 0 must not be in the domain of this expression.
.................................................
......!!!!!!!!...................................
09:32:25 ** The denominator of this expression cannot be zero, since division by zero is undefined. Since x^3 + x factors into (x^2 + 1) ( x ) we see that x^3 + x = 0 only if x^2 + 1 = 0 or x = 0. Since x^2 cannot be negative x^2 + 1 cannot be 0, so x = 0 is indeed the only value for which x^3 + x = 0. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:33:38 query R.2.73 (was R.4.6). What is (-4)^-2 and how did you use the laws of exponents to get your result?
......!!!!!!!!...................................
RESPONSE --> -4 ^2= -4*-4= 16. A negative number multiplied by itself= a positive
.................................................
......!!!!!!!!...................................
09:34:50 **Since a^-b = 1 / (a^b), we have (-4)^-2 = 1 / (-4)^2 = 1 / 16. **
......!!!!!!!!...................................
RESPONSE --> I messed that up completely- okay. So (-4)^2= 1/a, a reciprocal. 1/4^2= 1/16. I should have remembered that!
.................................................
......!!!!!!!!...................................
09:37:42 query Extra Problem. What is (3^-2 * 5^3) / (3^2 * 5) and how did you use the laws of exponents to get your result?
......!!!!!!!!...................................
RESPONSE --> 3^2=9* 5^3=75= 675/3^2=9*5=45 675/45= 15. Compute exponents first and then multiply.
.................................................
......!!!!!!!!...................................
09:41:38 ** (3^(-2)*5^3)/(3^2*5). Grouping factors with like bases we have 3^(-2)/3^2 * 5^3 / 5. Using the fact that a^b / a^c = a^(b-c) we get 3^(-2 -2) * 5^(3-1), which gives us 3^-4 * 5^2. Using a^(-b) = 1 / a^b we get (1/3^4) * 5^2. Simplifying we have (1/81) * 25 = 25/81. **
......!!!!!!!!...................................
RESPONSE --> Again, my mistake. Somehow I missed a negative number (2) in this equation. So, I group the factors that have like bases together (3), subtract the exponents= 3^(-2-2)* 5^3-1/5. 3^-4 would be a reciprocal= 1/81. 5^3-1= 5^2=25 *1/81=25/81. Now I understand where I went wrong.
.................................................
......!!!!!!!!...................................
09:51:07 query R.2.94. Express [ 5 x^-2 / (6 y^-2) ] ^ -3 with only positive exponents and explain how you used the laws of exponents to get your result.
......!!!!!!!!...................................
RESPONSE --> [ 5x^-2/(6y^-2)] ^-3= 5x^-2^-3= 5x^6/ (6y^-2)^-3=5x^6/1/6y^6=30x^6/y^6. In the laws of exponents, a number raised to a power then raised to another power= multiply the powers. In the case of a number with a negative exponent, it is equal to the reciprocal of the number raised to the same power, now positive.
.................................................
......!!!!!!!!...................................
09:52:15 [ 5 x^-2 / (6 y^-2) ] ^ -3 = (5 x^-2)^-3 / (6 y^-2)^-3, since (a/b)^c = a^c / b^c. This simplifies to 5^-3 (x^-2)^-3 / [ 6^-3 (y^-2)^-3 ] since (ab)^c = a^c b^c. Then since (a^b)^c = a^(bc) we have 5^-3 x^6 / [ 6^-3 y^6 ] . We rearrange this to get the result 6^3 x^6 / (5^3 y^6), since a^-b = 1 / a^b.
......!!!!!!!!...................................
RESPONSE --> I see the answer, but I have to now do this on paper.
.................................................
......!!!!!!!!...................................
09:54:49 query Extra Problem. Express (-8 x^3) ^ -2 with only positive exponents and explain how you used the laws of exponents to get your result.
......!!!!!!!!...................................
RESPONSE --> (-8x^3)^-2= 1/64x^6. A neagtive exponent= reciprocal.
.................................................
......!!!!!!!!...................................
09:55:14 ** ERRONEOUS STUDENT SOLUTION: (-8x^3)^-2 -1/(-8^2 * x^3+2) 1/64x^5 INSTRUCTOR COMMENT:1/64x^5 means 1 / 64 * x^5 = x^5 / 64. This is not what you meant but it is the only correct interpretation of what you wrote. Also it's not x^3 * x^2, which would be x^5, but (x^3)^2. There are several ways to get the solution. Two ways are shown below. They make more sense if you write them out in standard notation. ONE CORRECT SOLUTION: (-8x^3)^-2 = (-8)^-2*(x^3)^-2 = 1 / (-8)^2 * 1 / (x^3)^2 = 1/64 * 1/x^6 = 1 / (64 x^5). Alternatively (-8 x^3)^-2 = 1 / [ (-8 x^3)^2] = 1 / [ (-8)^2 (x^3)^2 ] = 1 / ( 64 x^6 ). **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
09:59:16 query R.2.90 (was R.4.36). Express (x^-2 y) / (x y^2) with only positive exponents and explain how you used the laws of exponents to get your result.
......!!!!!!!!...................................
RESPONSE --> (x^-2 y)/x y^2)= xy^2 *y*1/x^2= xy^3/x^2= y^3/x
.................................................
......!!!!!!!!...................................
10:00:04 ** (1/x^2 * y) / (x * y^2) = (1/x^2 * y) * 1 / (x * y^2) = y * 1 / ( x^2 * x * y^2) = y / (x^3 y^2) = 1 / (x^3 y). Alternatively, or as a check, you could use exponents on term as follows: (x^-2y)/(xy^2) = x^-2 * y * x^-1 * y^-2 = x^(-2 - 1) * y^(1 - 2) = x^-3 y^-1 = 1 / (x^3 y).**
......!!!!!!!!...................................
RESPONSE --> I do not understand this the first way it is written. I do understand the check method much better.
.................................................
......!!!!!!!!...................................
10:00:23 08-30-2005 10:00:23 query Extra Problem. . Express 4 x^-2 (y z)^-1 / [ (-5)^2 x^4 y^2 z^-5 ] with only positive exponents and explain how you used the laws of exponents to get your result.
......!!!!!!!!...................................
NOTES ------->
.................................................oٶpIیY
......!!!!!!!!...................................
11:00:10 query Extra Problem. . Express 4 x^-2 (y z)^-1 / [ (-5)^2 x^4 y^2 z^-5 ] with only positive exponents and explain how you used the laws of exponents to get your result.
......!!!!!!!!...................................
RESPONSE --> 4x^-2(yz)^-1= 1/16x^2 yz/ -5^2x^4y^2z^-5= 1/16x^2 yz* 25x^4 y^2 * 1/z^5= Reduce like variables= 25y/16 z^6.
.................................................
......!!!!!!!!...................................
11:08:42 ** Starting with 4x^-2(yz)^-1/ [ (-5)^2 x^4 y^2 z^-5] Squaring the -5 and using the fact that (yz)^-1 = y^1 * z^-1: 4x^-2 * y^-1 * z^-1/ [25 * x^4 * y^2 * z^-5} Grouping the numbers, and the x, the y and the z expression: (4/25) * (x^-2/x^4) * (y^-1/y^2) * (z^-1/z^-5) Simplifying by the laws of exponents: (4/25) * x^(-2-4) * y^(-1-2) * z^(-1+5) Simplifying further: (4/25) * x^-6 * y^-3 * z^4 Writing with positive exponents: 4z^4/ (25x^6 * y^3 ) **
......!!!!!!!!...................................
RESPONSE --> I am having bad difficulty with the laws of exponents. This problem especially. I can now understand the grouping and simplification with the laws of exponents. I cannot get to the step where the answer has been written with only positive exponents- I thought if it had a negative exponent, it was equal to a reciprocal?! 1/a^n. Okay, let me try from the last step I understand: (4/25)* 1/x^6*1/y^3 *z^4/1. Multiply straight across, top and bottom separately. That yields 4z^4/ 25x^6*y^3. Okay, I understand it now, so maybe this is not a law of exponents issue, but a grouping issue I am having. I will review this again and work at it.
.................................................
......!!!!!!!!...................................
11:09:53 query R.2.122 (was R.4.72). Express 0.00421 in scientific notation.
......!!!!!!!!...................................
RESPONSE --> 4.21*10^-3
.................................................
......!!!!!!!!...................................
11:10:03 ** 0.00421 in scientific notation is 4.21*10^-3. This is expressed on many calculators as 4.21 E-4. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
11:10:35 query R.2.128 (was R.4.78). Express 9.7 * 10^3 in decimal notation.
......!!!!!!!!...................................
RESPONSE --> 9700
.................................................
......!!!!!!!!...................................
11:10:41 ** 9.7*10^3 in decimal notation is 9.7 * 1000 = 9700 **
......!!!!!!!!...................................
RESPONSE --> okay
.................................................
......!!!!!!!!...................................
11:15:45 query R.2.150 (was R.2.78) If an unhealthy temperature is one for which | T - 98.6 | > 1.5, then how do you show that T = 97 and T = 100 are unhealthy?
......!!!!!!!!...................................
RESPONSE --> Substitute the alternate values of T into the statement. If T=100- 98.6, then T= 1.4 degrees. 1.4 however is not greater than 1.5. Then substitute 97 in for T= 97-98.6= -1.6 degrees. -1.6 degrees is not greater than 1.5 degrees either. Neither 97 degrees nor 100 degrees are unhealthy temperatures.
.................................................
......!!!!!!!!...................................
11:16:27 ** You can show that T=97 is unhealthy by substituting 97 for T to get | -1.6| > 1.5, equivalent to the true statement 1.6>1.5. But you can't show that T=100 is unhealthy, when you sustitute for T then it becomes | 100 - 98.6 | > 1.5, or | 1.4 | > 1.5, giving us 1.4>1.5, which is an untrue statement. **
......!!!!!!!!...................................
RESPONSE --> I forgot to take the absolute value of the -1.6 degrees. Otherwise, I got this right.
.................................................
}aʎzݥC assignment #003 C{ꃌtKޡY College Algebra 08-30-2005
......!!!!!!!!...................................
11:19:34 query R.3.12 (was R.3.6) What is the hypotenuse of a right triangle with legs 14 and 48 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> Pythagorean Theorem= a^2+b^2=c^2 14^2+48^2=c^2 196+2304=2500 2500=c^2 sqrt 2500=c c=50
.................................................
......!!!!!!!!...................................
11:19:48 ** The Pythagorean Theorem tells us that c^2 = a^2 + b^2, where a and b are the legs and c the hypotenuse. Substituting 14 and 48 for a and b we get c^2 = 14^2 + 48^2, so that c^2 = 196 + 2304 or c^2 = 2500. This tells us that c = + sqrt(2500) or -sqrt(2500). Since the length of a side can't be negative we conclude that c = +sqrt(2500) = 50. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
11:21:50 query R.3.18 (was R.3.12). Is a triangle with legs of 10, 24 and 26 a right triangle, and how did you arrive at your answer?
......!!!!!!!!...................................
RESPONSE --> Pythagorean Theorem: a^2+b^2=c^2 10^2+24^2=26^2 100+576=676 676 does = 676 This is a right triangle.
.................................................
......!!!!!!!!...................................
11:21:56 ** Using the Pythagorean Theorem we have c^2 = a^2 + b^2, if and only if the triangle is a right triangle. Substituting we get 26^2 = 10^2 + 24^2, or 676 = 100 + 576 so that 676 = 676 This confirms that the Pythagorean Theorem applies and we have a right triangle. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
11:30:44 query R.3.30 (was R.3.24). What are the volume and surface area of a sphere with radius 3 meters, and how did you obtain your result?
......!!!!!!!!...................................
RESPONSE --> Surface area of a sphere is =4*pi*radius length^2 Volume of a sphere=4/3*pi*length of the radius^3 S=4pi(3)^2=4 pi (9)=36 m squared*pi V=4/3*pi*(3)^3 V=4/3*pi*27m cubed V=36 m cubed*pi
.................................................
......!!!!!!!!...................................
11:31:52 ** To find the volume and surface are a sphere we use the given formulas: Volume = 4/3 * pi * r^3 V = 4/3 * pi * 3^3 V = 4/3 * pi * 27 V = 36pi m^3 Surface Area = 4 * pi * r^2 S = 4 * pi * 3^2 S = 4 * pi * 9 S = 36pi m^2. **
......!!!!!!!!...................................
RESPONSE --> I wrote the answers a little out of order, and I wasn't sure about using the ^ symbol in an answer, but I got it now!
.................................................
......!!!!!!!!...................................
11:32:14 08-30-2005 11:32:14 query R.3.42 (was R.3.36). A pool of radius 10 ft is enclosed by a deck of width 3 feet. What is the area of the deck and how did you obtain this result?
......!!!!!!!!...................................
NOTES ------->
.................................................Ilj{xwȰ
assignment #003 C{ꃌtKޡY College Algebra 08-30-2005
......!!!!!!!!...................................
12:10:48 query R.3.12 (was R.3.6) What is the hypotenuse of a right triangle with legs 14 and 48 and how did you get your result?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:10:50 ** The Pythagorean Theorem tells us that c^2 = a^2 + b^2, where a and b are the legs and c the hypotenuse. Substituting 14 and 48 for a and b we get c^2 = 14^2 + 48^2, so that c^2 = 196 + 2304 or c^2 = 2500. This tells us that c = + sqrt(2500) or -sqrt(2500). Since the length of a side can't be negative we conclude that c = +sqrt(2500) = 50. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:10:52 query R.3.18 (was R.3.12). Is a triangle with legs of 10, 24 and 26 a right triangle, and how did you arrive at your answer?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:10:54 ** Using the Pythagorean Theorem we have c^2 = a^2 + b^2, if and only if the triangle is a right triangle. Substituting we get 26^2 = 10^2 + 24^2, or 676 = 100 + 576 so that 676 = 676 This confirms that the Pythagorean Theorem applies and we have a right triangle. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:10:55 query R.3.30 (was R.3.24). What are the volume and surface area of a sphere with radius 3 meters, and how did you obtain your result?
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:11:04 ** To find the volume and surface are a sphere we use the given formulas: Volume = 4/3 * pi * r^3 V = 4/3 * pi * 3^3 V = 4/3 * pi * 27 V = 36pi m^3 Surface Area = 4 * pi * r^2 S = 4 * pi * 3^2 S = 4 * pi * 9 S = 36pi m^2. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
12:14:14 query R.3.42 (was R.3.36). A pool of radius 10 ft is enclosed by a deck of width 3 feet. What is the area of the deck and how did you obtain this result?
......!!!!!!!!...................................
RESPONSE --> If the pool has a radius of 10 ft, then its diameter= 20 feet (d=2r). If the width of the deck all the way around the pool is 3 feet, then the diameter of the pool plus 3 feet on either side= the width of the deck= 26 ft. The area of the deck= length of the deck *width of the deck. 26 ft*26ft= 676 ft^2
.................................................
......!!!!!!!!...................................
12:18:10 ** The deck plus the pool gives you a circle of radius 10 ft + 3 ft = 13 ft. The area of the deck plus the pool is therefore pi * (13 ft)^2 = 169 pi ft^2. So the area of the deck must be deck area = area of deck and pool - area of pool = 169 pi ft^2 - 100 pi ft^2 = 69 pi ft^2. **
......!!!!!!!!...................................
RESPONSE --> I messed up and assumed the deck was square. Okay, so the radius of the circle plus the width of the deck= 13 feet. A= pi* r^2= A= pi* (13ft)^2= A= 169 pi ft^2 The area of the pool only= pi* (10ft)^2= 100 pi ft^2 169 pi ft^2-100 pi ft^2= area of deck only= 69 pi ft ^2. Now I understand.
.................................................
......!!!!!!!!...................................
12:22:18 Query R.4.36 (was R.5.30). What is the single polynomial that is equal to 8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 )?
......!!!!!!!!...................................
RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. Always critique your solutions by describing any insights you had or errors you makde, and by explaining how you can make use of the insight or how you now know how to avoid certain errors. Also pose for the instructor any question or questions that you have related to the problem or series of problems. 32 x^3 - 24x^2 - 8- (24 x^3+48x-12) Change to addition and change signs within second expression= 32 x^3-24x^2-8+ -24x^3 -48 x+12= 8x^3-24x^2-48x-4
.................................................
......!!!!!!!!...................................
12:22:35 ** ERRONEOUS STUDENT SOLUTION: To make this problem into a single polynomial, you can group like terms together. (8-6)+ (4x^3-4x^3) + (-3x^2) + (8x) + (-1+2). Then solve from what you just grouped...2 (-3x^2+8x+1). INSTRUCTOR CORRECTION: 8 is multiplied by the first polynomial and 6 by the second. You can't isolate them like that. Starting with 8 ( 4 x^3 - 3 x^2 - 1 ) - 6 ( 4 x^3 + 8 x - 2 ) use the Distributive Law to get 32 x^3 - 24 x^2 - 8 - 24 x^3 - 48 x + 12. Then add like terms to get 8x^3 - 24x^2 - 48x + 4 **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
12:28:36 Query R.4.60 (was R.5.54). What is the product (-2x - 3) ( 3 - x)?
......!!!!!!!!...................................
RESPONSE --> Multiply each item in the second parenthesis by one item from the first set of parentheses. -2x(3-x)=- 6x+2x^2 -3(3-x)= -9+3x Add the two answers: 2x^2+6x+3x-9 = 2x^2+9x-9
.................................................
......!!!!!!!!...................................
12:29:25 ** Many students like to use FOIL but it's much better to use the Distributive Law, which will later be applied to longer and more complicated expressions where FOIL does not help a bit. Starting with (-2x - 3) ( 3 - x) apply the Distributive Law to get -2x ( 3 - x) - 3 ( 3 - x). Then apply the Distributive Law again to get -2x(3) - 2x(-x) - 3 * 3 - 3 ( -x) and simiplify to get -6x + 2 x^2 - 9 + 3x. Add like terms to get 2 x^2 - 3 x - 9. **
......!!!!!!!!...................................
RESPONSE --> I had a problem with a positive sign in my equation, but I see it now.
.................................................
......!!!!!!!!...................................
12:31:09 Query R.4.66 (was R.5.60). What is the product (x - 1) ( x + 1) and how did you obtain your result using a special product formula?
......!!!!!!!!...................................
RESPONSE --> This is the difference of two squares. This would be equal to x^2- (1)^2
.................................................
......!!!!!!!!...................................
12:32:27 ** Starting with (x-1)(x+1) use the Distributive Law once to get x ( x + 1) - 1 ( x+1) then use the Distributive Law again to get x*x + x * 1 - 1 * x - 1 * 1. Simplify to get x^2 +- x - x + - 1. Add like terms to get x^2 - 1. **
......!!!!!!!!...................................
RESPONSE --> I am not sure why the difference of two squares rule did not apply, but I do see how to use the distributive law to get the answer.
.................................................
......!!!!!!!!...................................
12:34:54 Query R.4.84 (was R.5.78). What is (2x + 3y)^2 and how did you obtain your result using a special product formula?
......!!!!!!!!...................................
RESPONSE --> Using FOIL: (2x+3y)(2x+3y)=4x^2+6xy+6xy+9y^2= 4x^2+12xy+9y^2
.................................................
......!!!!!!!!...................................
12:35:32 ** The Special Product is (a + b)^2 = a^2 + 2 a b + b^2. Letting a = 2x and b = 3y we get (2x)^2 + 2 * (2x) * (3y) + (3y)^2, which we expand to get 4 x^2 + 12 x y + 9 y^2. **
......!!!!!!!!...................................
RESPONSE --> I used FOIL and got the same thing, but I do understand the special product formula.
.................................................
......!!!!!!!!...................................
12:37:59 Query R.4.90 (was R.5.102). Explain why the degree of the product of two polynomials equals the sum of their degrees.
......!!!!!!!!...................................
RESPONSE --> According to the Laws of Exponents, when you multiply two polynomials, you add the exponents of the like variables.
.................................................
......!!!!!!!!...................................
12:38:19 ** STUDENT ANSWER AND INSTRUCTOR COMMENTS: The degree of the product of two polynomials equals the sum of their degrees because you use the law of exponenents and the ditributive property. INSTRUCOTR COMMENTS: Not bad. A more detailed explanation: The Distributive Law ensures that you will be multiplying the highest-power term in the first polynomial by the highest-power term in the second. Since the degree of each polynomial is the highest power present, and since the product of two powers gives you an exponent equal to the sum of those powers, the highest power in the product will be the sum of the degrees of the two polynomials. Since the highest power present in the product is the degree of the product, the degree of the product is the sum of the degrees of the polynomials. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
12:38:43 Query Add comments on any surprises or insights you experienced as a result of this assignment.
......!!!!!!!!...................................
RESPONSE --> I really need to work on grouping and memorizing what Law of Exponent applies and when.
.................................................
......!!!!!!!!...................................
12:39:37 08-30-2005 12:39:37 R.5.22 (was R.6.18). What do you get when you factor 36 x^2 - 9 and how did you get your result?
......!!!!!!!!...................................
NOTES -------> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. Always critique your solutions by describing any insights you had or errors you makde, and by explaining how you can make use of the insight or how you now know how to avoid certain errors. Also pose for the instructor any question or questions that you have related to the problem or series of problems.
.................................................ҧB·ϟՙي]
ǢWڮVKwz| assignment #006 C{ꃌtKޡY College Algebra 08-30-2005
......!!!!!!!!...................................
12:46:29 R.5.22 (was R.6.18). What do you get when you factor 36 x^2 - 9 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> This is the difference of perfect squares (6x-3)(6x+3)
.................................................
......!!!!!!!!...................................
12:46:34 ** 36x^2-9 is the difference of two squares. We write this as (6x)^2-3^2 then get (6x-3)(6x+3), using the special formula difference of two squars. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
12:47:21 R.5.28 (was R.6.24 What do you get when you factor x^2 + 10 x + 1 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> This has to be a prime polynomial
.................................................
......!!!!!!!!...................................
12:47:51 ** STUDENT SOLUTION: x^2+10x+1 is prime because there are no integers whose product is 10 and sum is 1 INSTRUCTOR COMMENTS: The sum should be 10 and the product 1. I agree that there are no two integers with this property. Furthermore there are no two rational numbers with this property. So you would never find the factors by inspection. However that doesn't mean that there aren't two irrational numbers with the property. For example 10 and 1/10 come close, with product 1 and sum 10.1. The quadratic formula tells you in fact that the two numbers are ( -10 + sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) and ( -10 - sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) . Since 10^2 - 4 = 96 is positive, these are real numbers, both irrational. So the polynomial isn't prime. **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
12:50:05 R.5.34 (was R.6.30). What do you get when you factor x^3 + 125 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> This is the sum of two cubes. Therefore x^3+125= (x+5)(x^2-5x+5)
.................................................
......!!!!!!!!...................................
12:50:31 ** x^3+125 is the sum of two cubes, with 125 = 5^3. We know that a^3 + b^3 = (a+b) ( a^2 - 2 a b + b^2). So we write x^3+5^3 = (x+5)(x^2-5x+25). **
......!!!!!!!!...................................
RESPONSE --> Other than forgetting to square the 5 in the second part of the factoring, I got it.
.................................................
......!!!!!!!!...................................
12:51:31 R.5.46 (was R.6.42). What do you get when you factor x^2 - 17 x + 16 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> (x-1)(x-16)
.................................................
......!!!!!!!!...................................
12:51:42 ** x^2-17x+16 is of the form (x + a) ( x + b) = x^2 + (a + b) x + ab, with a+b = -17 and ab = 16. If ab = 16 then we might have a = 1, b = 16, or a = 2, b = 8, or a = -2, b = -8, or a = 4, b = 4, or a = -1, b = -16, or a = -4, b = -4. These are the only possible integer factors of 16. In order to get a + b = -17 we must have at least one negative factor. The only possibility that gives us a + b = -17 is a = -1, b = -16. So we conclude that x^2 - 17 x + 16 = (x-16)(x-1). **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
12:59:10 R.5.52 (was R.6.48). What do you get when you factor 3 x^2 - 3 x + 2 x - 2 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> Factor by grouping= (3x^2-3x) and (2x-2) 3x(x-1) + 2(x-1)= (3x+2)(x-1)
.................................................
......!!!!!!!!...................................
12:59:16 ** This expression can be factored by grouping: 3x^2-3x+2x-2 = (3x^2-3x)+(2x-2) = 3x(x-1)+2(x-1) = (3x+2)(x-1). **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
13:01:29 R.5.64 (was R.6.60). What do you get when you factor 3 x^2 - 10 x + 8 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> I figured out that 8= 2*4 or -2*-4 (3x-4)(x-2)= 3x^2-10x+8
.................................................
......!!!!!!!!...................................
13:01:41 ** Possibilities are (3x - 8) ( x - 1), (3x - 1) ( x - 8), (3x - 2) ( x - 4), (3x - 4) ( x - 2). The possibility that gives us 3 x^2 - 10 x + 8 is (3x - 4) ( x - 2). **
......!!!!!!!!...................................
RESPONSE --> ok
.................................................
......!!!!!!!!...................................
13:02:49 ** This expression factors, but not into binomtials with integer coefficients. We could list all the possibilities: (x + 7) ( -x + 2), (x + 2) ( -x + 7), (x + 14) ( -x + 1), (x + 1)(-x + 14), but none of these will give us the desired result. For future reference: You won't find the factors in the usual manner. The quadratic formula tells us that there are factors ( -6 + sqrt(6^2 - 4 * 14 * (-1) )) / (2 * -1) and ( -6 - sqrt(6^2 - 4 * 14 * (-1) ) ) / (2 * -1) . Since sqrt(6^2 - 4 * 14 * (-1) ) = sqrt(36 + 56) = sqrt(92) is a real number these solutions are real numbers but again, as in a previous example, they aren't rational numbers and nobody could ever find them by inspection. This is not something you're expected to do at this point. **
......!!!!!!!!...................................
RESPONSE --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. Always critique your solutions by describing any insights you had or errors you makde, and by explaining how you can make use of the insight or how you now know how to avoid certain errors. Also pose for the instructor any question or questions that you have related to the problem or series of problems. ok, I am glad you don't expect me to do this at this point. I am not familiar at all with quadratic formulas.
.................................................
}˵cܯwĤ愬 assignment #007 C{ꃌtKޡY College Algebra 08-30-2005