1. 4.6 Reasoning about Factoring Polynomials
AREA
= length x width
= (3x – 2)(2x + 1)
= 6x2 + 3x – 4x – 2
= 6x2 – x – 2
Thus, the area of the rectangle is represented by the trinomial x2 + -x – 2.
Recall:
2x + 1
3x - 2

2. Volume
Now we’ll see how the trinomial 8x3-6x2 – 5x represents the volume of a rectangular prism.
How would we find the dimensions of the rectangular prism?

3. Volume V = l x w x h and V = 8x3 - 6x2 – 5x We can factor out an ‘x’:
To factor the trinomial, use decomposition:
Find 2 numbers whose sum is -6 and whose product is (8)(-5) = -40. They are -10 and 4. Use these numbers to decompose the middle term.
V = x(8x2 – 10x + 4x – 5)
Notice that there is a common factor of (4x-5)
V = x(2x(4x-5) + (4x-5))
Now using a trick we learned earlier…

4. Volume
V = x(2x(4x-5) + (4x-5)) = x(2x + 1)(4x – 5) Thus, possible dimensions of the rectangular prism are V = (x)*(2x+1)*(4x-5)

5. Example #2 Factor the expressions x2 + x – 132
What 2 numbers multiply to -132 and add to +1?
12 and -11
So, x2 + x – 132 = (x + 12)(x – 11)

6. Example #2 cont’d
b) 16x2 – 88x Recognize that 16 and 121 are perfect squares. When you double the product of their square roots, you get the middle term. 16 𝑥 2 =4𝑥; 121 =11 So, 16x2 – 88x + 121= (4x - 11)2 Notice that it’s (4x – 11)2 not (4x + 11)2. Remember that it is minus because the middle term in the trinomial is negative.

7. Example #2 cont’d c) -18x4 + 32x2
First, factor out whatever you can to simplify:
=-2x2(9x2 - 16)
Now we recognize the item in brackets as a difference of squares!
(9x2 – 16): 9 𝑥 2 =3𝑥 and 16 =4, so:
(9x2 – 16) = (3x + 4)(3x – 4)
Overall, -18x4 + 32x2 = -2x2(3x+4)(3x-4)

8. Example #3 Factor x5y + x2y3 – x3y3 – y5
Since there is a ‘y’ in each term, factor it out:
= y(x5 + x2y2 – x3y2 – y4)
We can take out x2 from the first 2 terms, and take out y2 from the second 2 terms:
= y(x2(x3 + y2) – y2(x3 – y2))
Now (x3 – y2) appears in both terms!
= y((x2 - y2)(x3 + y2))
= y(x2 - y2)(x3 + y2)
Overall, x5y + x2y3 – x3y3 – y5= y(x2 - y2)(x3 + y2)

9. Example #3 cont’d If you want to take it a step further…
x5y + x2y3 – x3y3 – y5= y(x2 - y2)(x3 + y2)
Notice, (x2 – y2) is a difference of squares, and as such is equal to (x + y)(x – y), so…
OVERALL:
x5y + x2y3 – x3y3 – y5= y(x + y)(x – y)(x3 + y2)