12.3: More Factoring with Diagrams


Let’s learn a way to divide polynomials.

12.1: Notice and Wonder: A Different Use for Diagrams
What do you notice? What do you wonder?

A. (x−3)(x+5)=x2+2x−15
x 5
x x2 5x
−3 −3x −15
B. (x−1)(x2+3x−4)=x3+2x2−7x+4
x2 3x −4
x x3 3x2 −4x
−1 −x2 −3x 4
C. (x−2)(?)=(x3−x2−4x+4)

x x3
−2
i notice how the numbers go in order and i wonder how all of the numbers because the same


12.2: Factoring with Diagrams
Priya wants to sketch a graph of the polynomial f defined by f(x)=x3+5x2+2x−8. She knows f(1)=0, so she suspects that (x−1) could be a factor of x3+5x2+2x−8 and writes (x3+5x2+2x−8)=(x−1)(?x2+?x+?) and draws a diagram.



1. Finish Priya’s diagram.

Type your answers in the boxes.


x x3
−1


2. Write f(x) as the product of (x−1) and another factor.

Type your answer in the box.

f(x)=


3. Write f(x) as the product of three linear factors.

Type your answer in the box.

f(x)=


4. Make a sketch of y=f(x).

Scribble

Blank x y coordinate plane, no grid. Horizontal axis from negative 10 to 8 by 2’s.
12.3: More Factoring with Diagrams
Here are some polynomial functions with known factors. Rewrite each polynomial as a product of linear factors. Note: you may not need to use all the columns in each diagram. For some problems, you may need to make another diagram.

1. A(x)=x3−7x2−16x+112, (x−7)
x2
x x3 0
−7 −7x2
Type your answer in the box.



A(x)=
2. B(x)=2x3−x2−27x+36, (x−32)


2x2
x 2x3 2x2
−32 −3x2
Type your answer in the box.



B(x)=
3. C(x)=x3−3x2−13x+15, (x+3)



x
3