#$&*
course Mth 158
6/9/12 0445am
If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution:
If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
006. `* 6
R.5.22 (was R.6.18). What do you get when you factor 36 x^2 - 9 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(6x)^2-3^2
Two square difference.
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
* * ** 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 squares. **
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Self-critique (if necessary):
I got the first step but I dont quite understand how u get the second part (6x-3)(6x+3), got any instructions for me???
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Self-critique Rating:
@&
It's a basic and frequently-encountered fact that
(a - b) ( a + b) = a^2 - b^2.
So any time you encounter the difference of two perfect squares, you can use this to factor the expression.
36 x^2 - 9 is, as you say
(6x)^2 - 3^2.
This expression is of the form
a^2 - b^2
with a = 6x and b = 3. So applying the formula
a^2 - b^2 = (a - b) ( a + b )
you get
(6x)^2 - 3^2 = (6x - 3) ( 6x + 3).
*@
2
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Question:
R.5.32 \ 28 (was R.6.24) What do you get when you factor 25 x^2 + 10 x + 1 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(5) x^2+10x+1
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1
.............................................
Given Solution:
* * ** STUDENT SOLUTION: x^2+10x+1 is prime because there are no integers whose product is 10 and sum is 1
INSTRUCTOR COMMENTS: The sum should be 10 and the product 1. I agree that there are no two integers with this property. Furthermore there are no two rational numbers with this property.
So you would never be able to find the factors by inspection.
However that doesn't mean that there aren't two irrational numbers with the property. For example 10 and 1/10 come close to the criteria, with product 1 and sum 10.1.
The quadratic formula tells you in fact that the two numbers are
( -10 + sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) and
( -10 - sqrt( 10^4 - 4 * 1 * 1) ) / (2 * 1) .
Since 10^2 - 4 = 96 is positive, these are real numbers, both irrational. So the polynomial isn't prime. **
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Self-critique (if necessary):
I still dont understand this one, even with the explanation given above, could you help explain it a little clearer to me???
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Self-critique Rating:
@&
Let's revisit this one when you get to the quadratic formula. That's actually something you're supposed to know when you start this course, but at this point most students don't.
*@
2
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Question:
R.5.34 (was R.6.30). What do you get when you factor x^3 + 125 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Sum of two cubes
125 = 5^3
a^3 + b^3 = (a+b) ( a^2 - 2 a b + b^2)
x^3+5^3 = (x+5)(x^2-2 (x) (5) +25)
x^3+5^3 = (x+5)(x^2 - 2x -10 +25)
x^3+5^3 = (x+5)(x^2 - 2x + 15)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
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).
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Self-critique (if necessary):
I got confused on the third step somehow with the a b part, I dont know how it became
-5x???
@&
Two of your steps are
x^3+5^3 = (x+5)(x^2-2 (x) (5) +25)
x^3+5^3 = (x+5)(x^2 - 2x -10 +25)
There is an error in going from the first of these steps to the second.
-2(x)(5) = -10 x, not -2x - 10.
My guess is that you were thinking distributive law.
It is true that
-2 ( x + 5) = -2 x - 10
but there is no + in the expression. -2 * x * 5 is pure multiplication. By order of operations multiplications are done in order so
-2 * x * 5 = (-2 * x) * 5 = (-2x) * 5
and
(-2x) * 5 = -10 x.
*@
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Self-critique Rating:
2
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Question:
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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(x + a) ( x + b) = x^2 + (a + b) x + ab a+b=-17 ab=16
x^2 - 17 x + 16 = (x-16)(x-1)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
* * ** x^2-17x+16 is of the form (x + a) ( x + b) = x^2 + (a + b) x + ab, with a+b = -17 and ab = 16.
If ab = 16 then, if a and b happen to be integers, we have the following possibilities:
a = 1, b = 16, or
a = 2, b = 8, or
a = -2, b = -8, or
a = 4, b = 4, or
a = -1, b = -16, or
a = -4, b = -4.
These are the only possible integer factors of 16.
In order to get a + b = -17 we must have at least one negative factor. So the possibilities are reduced to
a = -2, b = -8, or
a = -1, b = -16, or
a = -4, b = -4.
The only of the these possibilities that gives us a + b = -17 is a = -1, b = -16. So we conclude that
x^2 - 17 x + 16 = (x-16)(x-1). **
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Self-critique (if necessary):
Im not at all sure how I got this. I was close to doing it like the above solution, but Im not 100% how I got it.
------------------------------------------------
Self-critique Rating:
@&
Try to break down the given solution step by step:
Do you understand why
(x + b) ( x + b) = x^2 + (a + b) x + ab?
So you understand why
x^2 - 17 x + 16
matches this form if a + b = -17 and a * b = 16?
So you understand if a b = 16, then the following are all possible:
a = 1, b = 16, or
a = 2, b = 8, or
a = -2, b = -8, or
a = 4, b = 4, or
a = -1, b = -16, or
a = -4, b = -4.
Do you understand that (a + b) = -17 is satisfied if a = -1 and b = -16, and by none of the other listed possibilities?
*@
#$&*
course Mth 158
6/9/12 0445am
If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
Your solution, attempt at solution:
If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
006. `* 6
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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(6x)^2-3^2
Two square difference.
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
* * ** 36x^2-9 is the difference of two squares. We write this as
(6x)^2-3^2
then get
(6x-3)(6x+3),
using the special formula difference of two squares. **
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
I got the first step but I dont quite understand how u get the second part (6x-3)(6x+3), got any instructions for me???
@&
It's a basic and frequently-encountered fact that
(a - b) ( a + b) = a^2 - b^2.
You need to remember this, and understand why it's so.
So any time you encounter the difference of two perfect squares, you can use this to factor the expression.
36 x^2 - 9 is, as you say
(6x)^2 - 3^2.
This expression is of the form
a^2 - b^2
with a = 6x and b = 3. So applying the formula
a^2 - b^2 = (a - b) ( a + b )
you get
(6x)^2 - 3^2 = (6x - 3) ( 6x + 3).
*@
------------------------------------------------
Self-critique Rating:
2
*********************************************
Question:
R.5.32 \ 28 (was R.6.24) What do you get when you factor 25 x^2 + 10 x + 1 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(5) x^2+10x+1
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1
.............................................
Given Solution:
* * ** 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 be able to 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 to the criteria, 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. **
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Self-critique (if necessary):
I still dont understand this one, even with the explanation given above, could you help explain it a little clearer to me???
@&
Let's revisit this one when you get to the quadratic formula. That's actually something you're supposed to know when you start this course, but at this point most students don't.
*@
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Self-critique Rating:
2
*********************************************
Question:
R.5.34 (was R.6.30). What do you get when you factor x^3 + 125 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Sum of two cubes
125 = 5^3
a^3 + b^3 = (a+b) ( a^2 - 2 a b + b^2)
x^3+5^3 = (x+5)(x^2-2 (x) (5) +25)
x^3+5^3 = (x+5)(x^2 - 2x -10 +25)
x^3+5^3 = (x+5)(x^2 - 2x + 15)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
x^3+125 is the sum of two cubes, with 125 = 5^3.
We know that a^3 + b^3 = (a+b) ( a^2 - 2 a b + b^2).
So we write
x^3+5^3 = (x+5)(x^2-5x+25).
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
I got confused on the third step somehow with the a b part, I dont know how it became
-5x???
@&
Two of your steps are
x^3+5^3 = (x+5)(x^2-2 (x) (5) +25)
x^3+5^3 = (x+5)(x^2 - 2x -10 +25)
There is an error in going from the first of these steps to the second.
-2(x)(5) = -10 x, not -2x - 10.
My guess is that you were thinking distributive law.
It is true that
-2 ( x + 5) = -2 x - 10
but there is no + in the expression. -2 * x * 5 is pure multiplication. By order of operations multiplications are done in order so
-2 * x * 5 = (-2 * x) * 5 = (-2x) * 5
and
(-2x) * 5 = -10 x.
*@
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Self-critique Rating:
2
*********************************************
Question:
R.5.46 (was R.6.42). What do you get when you factor x^2 - 17 x + 16 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(x + a) ( x + b) = x^2 + (a + b) x + ab a+b=-17 ab=16
x^2 - 17 x + 16 = (x-16)(x-1)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
* * ** 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, if a and b happen to be integers, we have the following possibilities:
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. So the possibilities are reduced to
a = -2, b = -8, or
a = -1, b = -16, or
a = -4, b = -4.
The only of the these possibilities that gives us a + b = -17 is a = -1, b = -16. So we conclude that
x^2 - 17 x + 16 = (x-16)(x-1). **
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Self-critique (if necessary):
Im not at all sure how I got this. I was close to doing it like the above solution, but Im not 100% how I got it.
Try to break down the given solution step by step:
Do you understand why
(x + b) ( x + b) = x^2 + (a + b) x + ab?
So you understand why
x^2 - 17 x + 16
matches this form if a + b = -17 and a * b = 16?
So you understand if a b = 16, then the following are all possible:
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.
Do you understand that (a + b) = -17 is satisfied if a = -1 and b = -16, and by none of the other listed possibilities?
------------------------------------------------
Self-critique Rating:
2
*********************************************
Question:
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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Distributive law
3x^2-3x+2x-2
(3x^2-3x)+(2x-2)
3x(x-1)+2(x-1)
(3x+2)(x-1)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
* * ** This expression can be factored by grouping:
3x^2-3x+2x-2 =
(3x^2-3x)+(2x-2) =
3x(x-1)+2(x-1) =
(3x+2)(x-1). **
ADDITIONAL EXPLANATION:
To see that
(3x^2-3x)+(2x-2) =
3x(x-1)+2(x-1)
apply the distributive law to each term in the second expression:
3x ( x - 1) = 3 x^2 - 3x, and
2 ( x - 1) = 2x - 2.
To see that
3x(x-1)+2(x-1) =
(3x+2)(x-1)
apply the distributive law as follows:
(3x + 2) ( x - 1) = 3x * (x - 1) + 2 * (x - 1).
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Self-critique (if necessary):
I think I got this right using the distributive law.
@&
This looks good.
*@
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Self-critique Rating:
2
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Question:
R.5.64 (was R.6.60). What do you get when you factor 3 x^2 - 10 x + 8 and how did you get your result?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(x + a) ( x + b) = x^2 + (a + b) x + ab a+b=-10 ab=8
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1, this one completely lost me???
.............................................
Given Solution:
* * ** 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). **
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?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
(x + a) ( x + b) = x^2 + (a + b) x + ab a+b=6 ab=14
@&
Try to break this down in a way similar to the earlier problem of this nature.
*@
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1, I got a little more of this one worked out on paper, but none of them matched up and when I checked it with the solution, where did the z1 and z2 come into play at???
.............................................
Given Solution:
* * ** This expression factors, but not into binomials 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 often cannot find the factors by factoring in the usual manner; however it is always possible to find the factors of a second-degree trinomial using the quadratic formula.
The quadratic formula applied to this problem tells us that the factors are (x - z1) * (x - z2), were z1 and z2 are the solutions to the equation 14 + 6 x - x^2 = 0. Using the formula we find that
z1 = ( -6 + sqrt(6^2 - 4 * 14 * (-1) )) / (2 * -1) and
z2 = ( -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. **
"
@&
As specified in the statement,
z1 and z2 are the solutions to the equation 14 + 6 x - x^2 = 0
but this is under the heading 'for future reference'.
If you understand it, that's great and it will save you time in the near future. If not, don't worry about it now; you can deal with it when you get to it.
*@
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
@&
Check out my notes. I'm glad to answer additional questions.
*@