course Mth 158 {?W????U????????assignment #006
......!!!!!!!!...................................
13:30:43 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 --> I can see that this is a difference of squares. To solve the problem, I took the square root of 36=6 and the square root of 9=3. Then I factored it out and got ( 6x+3) ( 6x - 3) confidence assessment: 3
.................................................
......!!!!!!!!...................................
13:31:45 ** 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 --> I knew that this was a difference of squares and I got the correct answer. self critique assessment: 3
.................................................
......!!!!!!!!...................................
13:34:08 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 problem cannot be factored because there are no two numbers when multiplied together equal 1 and when added together equal 10. confidence assessment: 2
.................................................
......!!!!!!!!...................................
13:36:18 ** 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 --> I thought that thee might be an irrational number that could be a solution. I based my answer on the fact that there were no rational numbers that could be solutions. self critique assessment: 2
.................................................
......!!!!!!!!...................................
13:42:25 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 --> I know that 5^3 = 125. I use x^3 and 5^3 to factor. (x+5) ( x^2 - 5 + 25). I got 25 by squaring 5. confidence assessment: 1
.................................................
......!!!!!!!!...................................
13:46:02 ** 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 --> I'm not sure how to get -5x because in ( a^2 - 2 a b+b^2) that would make -5x, -10x if you multiplied 5*2*a. self critique assessment: 2
.................................................
......!!!!!!!!...................................
13:48:50 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 --> When I factored this I got (x-16) (x-1). I first thought of two numbers when multiplied together would equal 16 and when added together would equal -17. So -16 * -1 is 16 and -16 - 1 is -17. confidence assessment: 2
.................................................
......!!!!!!!!...................................
13:49:39 ** 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 --> That is what I did, I thought of two possible numbers that would work here and found the correct answer. self critique assessment: 2
.................................................
......!!!!!!!!...................................
14:04:07 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 --> I factored by grouping. I first grouped the first two terms together (3x^2 - 3) and found a common factor of 3x, which gave me 3x(x -1). Then I grouped the last two terms together (2x - 2) and found a common factor of 2, that gave me 2(x-1). The final step to put the common factors together. (3x + 2) ( x -1) confidence assessment: 2
.................................................
......!!!!!!!!...................................
14:04:23 ** 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 --> Thats what I did. self critique assessment: 3
.................................................
......!!!!!!!!...................................
14:08:08 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 first found the product of 3*8=24. Then I determined that 6*4 gave me 24 and 6+4 gave me 10. These are the factors I used to find ( 3x-6) ( x+4). confidence assessment: 2
.................................................
......!!!!!!!!...................................
14:10:16 ** 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 --> I messed up on that one. I multiplied wrong. I can see how to get the correct answer. self critique assessment: 2
.................................................
......!!!!!!!!...................................
14:13:00 R.5.82 (was R.6.78). What do you get when you factor 14 + 6 x - x^2 and how did you get your result?
......!!!!!!!!...................................
RESPONSE --> I rearranged the problem as -x^2 + 6x +14. This polynomial cannot be factored because there are no two rational numbers that when multiplied equal 14 and when added equal 6. confidence assessment: 1
.................................................
......!!!!!!!!...................................
14:14:50 ** 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 --> I could see that there were no possible solutions that were rational. I understand that the quadratic formula would give us irrational answers. self critique assessment: 2
................................................."