Recall:

4.6 Reasoning about Factoring Polynomials

2x + 1

3x - 2

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 6x2 + -x – 2.

Now we’ll see how the trinomial 8x3-6x2 – 5x represents the volume of a rectangular prism.

Volume

How would we find the dimensions of the rectangular prism?

V = l x w x h and V = 8x3 - 6x2 – 5x 1. We can factor out an ‘x’:

1. V = x(8x2 – 6x – 5)2. To factor the trinomial, use decomposition:

1. 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.

2. V = x(8x2 – 10x + 4x – 5)1. Notice that there is a common factor of (4x-5)

3. V = x(2x(4x-5) + (4x-5))1. Now using a trick we learned earlier…

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)

Volume

Factor the expressionsa) x2 + x – 132What 2 numbers multiply to -132 and add to +1?

12 and -11So, x2 + x – 132 = (x + 12)(x – 11)

Example #2

b) 16x2 – 88x + 121Recognize that 16 and 121 are perfect squares. When you double the product of their square roots, you get the middle term.

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.

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): and so:(9x2 – 16) = (3x + 4)(3x – 4)Overall, -18x4 + 32x2 = -2x2(3x+4)(3x-4)

Example #2 cont’d

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)

Example #3

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)

Example #3 cont’d