a Find factors or terms a and b, such that ab=- 12x^2 and a+b=- 4x.
B
b Find factors or terms a and b, such that ab=4x^2 and a+b=4x.
C
c Find factors or terms a and b, such that ab=- 10x^2 and a+b=- 9x.
D
d Find factors or terms a and b, such that ab=- 24x^2 and a+b= 10x.
A
a (x-6)(x+2)
B
b (2x+1)^2
C
c (2x+1)(x-5)
D
d (3x-2)(x+4)
Practice makes perfect
a Using the process from Problem 8-14, we know that x^2 and - 12 goes into the lower left and upper right corners of the generic rectangle.
To fill in the remaining two rectangles we need two x-terms that have a sum of - 4x and a product of - 12x^2.
Notice that the product is negative, which means one factor must be negative and the other must be positive. With this in mind, we factor - 12x^2 in as many ways as we can and then add the factors. When we find factors with a sum of - 4x, we have factored correctly.
|c|c|c|r|c|
[-1em]
Product & ax(bx) & ax+bx & Sum & - 4x? [0.2em]
[-1em]
- 12x^2 & - x(12x) & - x+12x& 11x & * [0.1em]
- 12x^2 & - 12x(x) & -12x+x& - 11x & * [0.1em]
- 12x^2 & - 3x(4x) & - 3x+4x& 1x & * [0.1em]
- 12x^2 & - 4x(3x) & - 4x+3x& - 1x & * [0.1em]
- 12x^2 & - 2x(6x) & - 2x+6x& 4x & * [0.1em]
- 12x^2 & - 6x(2x) & - 6x+2x& - 4x & âś“ [0.1em]
When one term is - 6x and the other is 2x, we have a product of - 12x^2 and a sum of - 4x. Now we can complete the diamond and generic rectangle.
To factor the quadratic expression we add each side of the generic rectangle and multiply the sums.
x^2-4x-12=(x+(- 6))(x+2)
⇓
x^2-4x-12=(x-6)(x+2)
b Like in Part A, we will create a generic rectangle and a diamond using the quadratic expression.
Both the product and the sum are positive. For this to be true, both factors and terms must be positive. With this in mind, let us factor 4x^2 in as many ways as we can and sum the factors.
|c|c|c|c|c|
[-1em]
Product & ax(bx) & ax+bx & Sum & 4x? [0.2em]
[-1em]
4x^2 & 4x(x) & 4x+x& 5x & * [0.1em]
4x^2 & 2x(2x) & 2x+2x& 4x & âś“ [0.1em]
When both factors are 2x we have a product of 4x^2 and a sum of 4x. Now we can complete the diamond and the generic rectangle.
To factor the quadratic expression we add the parts along each side of the generic rectangle and multiply the sums.
4x^2+4x+1=(2x+1)(2x+1)
⇓
4x^2+4x+1=(2x+1)^2
c Like in Parts A and B, we will create a generic rectangle and a diamond using the quadratic expression.
Notice that the product is negative, which means one factor must be negative and the other must be positive. With this in mind, let us factor - 10x^2 in as many ways as we can and add the factors.
|c|c|c|r|c|
[-1em]
Product & ax(bx) & ax+bx & Sum & - 9x? [0.2em]
[-1em]
- 10x^2 & - x(10x) & - x+10x& 9x & * [0.1em]
- 10x^2 & - 10x(x) & -10x+x & - 9x & âś“ [0.1em]
- 10x^2 & - 2x(5x) & -2x+5x& 3x & * [0.1em]
- 10x^2 & - 5x(2x) & -5x+2x& - 3x & * [0.1em]
When one factor is - 10x and the other is x, we have a product of - 10x^2 and a sum of - 9x. Now we can complete the diamond and generic rectangle.
To factor the quadratic expression we add the parts along each side of the generic rectangle and multiply the sums.
2x^2-9x-5=(2x+1)(x+(- 5))
⇓
2x^2-9x-5=(2x+1)(x-5)
d Like in previous parts, we will create a generic rectangle and a diamond using the quadratic expression.
Notice that the product is negative, which means one factor must be negative and the other positive. With this in mind, let us factor - 24x^2 in as many ways as we can and sum the factors.
|c|c|c|r|c|
[-1em]
Product & ax(bx) & ax+bx & Sum & 10x? [0.2em]
[-1em]
- 24x^2 & - 24x(x) & - 24x+x& - 23x & * [0.1em]
- 24x^2 & - x(24x) & - x+24x& 23x & * [0.1em]
- 24x^2 & - 12x(2x) & - 12x+2x& - 10x & * [0.1em]
- 24x^2 & - 2x(12x) & - 2x+12x& 10x & âś“ [0.1em]
- 24x^2 & - 8x(3x) & - 8x+3x& - 5x & * [0.1em]
- 24x^2 & - 3x(8x) & - 3x+8x& 5x & * [0.1em]
- 24x^2 & - 6x(4x) & - 6x+4x& - 2x & * [0.1em]
- 24x^2 & - 4x(6x) & - 4x+6x& 2x & * [0.1em]
When one factor is - 2x and the other is 12x, we have a product of -24x^2 and a sum of 10x. Now we can complete the diamond and generic rectangle.
To factor the quadratic expression we add the parts along each side of the generic rectangle and multiply the sums.
3x^2+10x-8=(3x+(-2))(x+4)
⇓
3x^2+10x-8=(3x-2)(x+4)
