Name _________________________
Date _______________
7.6 – Factoring Polynomials
Big Idea: You already know that there is a relationship between the factored form of a quadratic
equation, and the roots and x-intercepts of that quadratic equation. In this lesson you will learn how to
write higher-degree polynomial equations in factored form when you know the roots of the equation.
You’ll also discover useful techniques for converting a polynomial in general form to factored form.

A 3rd-Degree polynomial function is called a cubic function. Verify with your calculator that the
cubic functions below are equivalent.
y  x3  9 x 2  26 x  24
and
y  ( x  2)( x  3)( x  4)
Example A: Write the cubic functions for the graphs below. Notice that both graphs have the same xintercepts. You will need to utilize the equation y  a( x  r1 )( x  r2 )( x  r3 ) . One way to solve for a is
to substitute the coordinates of one other point, such as the y-intercept, into the function.
Roots: x =
Equation:
Roots: x =
y  a( x  ___)( x  ___)( x  ___)
Y-Intercept: ( _____ , _____ )
y  ___( x  ___)( x  ___)( x  ___)
Equation:
y  a( x  ___)( x  ___)( x  ___)
Y-Intercept: ( _____ , _____ )
y  ___( x  ___)( x  ___)( x  ___)

The factored form of a polynomial function tells you the zeros of the function and the xintercepts of the graph of the function. Recall that zeros are solutions to the equation
f ( x)  0 . One strategy for finding the real solutions of a polynomial is factoring. However,
not all polynomials can be factored.
Example B: Sketch the graph of each function using a graphing calculator, then determine its
factored form. (This example gives two different polynomials that CAN be factored)
y  x2  x  2
a.
b.
y  4 x3  8x 2  36 x  72
Roots:
Roots:
Equation in Factored Form:
Equation in Factored Form:
In Example B, you converted a function from general form to factored form by using a graph and
looking for the x-intercepts. This method works especially well when the zeros are integer values.
Once you know the zeros of a polynomial function, r1 , r2 , r3 , and so on, you can write the factored
form,
y  a( x  r1 )( x  r2 )( x  r3 ) …
You can also write a polynomial function in factored form when the zeros are not integers, or even
when they are nonreal.
Polynomials with real coefficients can be separated into three types:
4
3
2
Note that the coefficients for y  ax  bx  cx  dx  e are a, b, c, d, and e.
o polynomials that can’t be completely factored with real numbers;
o polynomials that can be factored with real numbers, but some of the roots are not “nice”
integer or fractional values;
o
polynomials that can be factored and have all integer or fractional roots. For example,
consider these cases of quadratic functions:
What happens when the graph of a quadratic function has exactly one point of intersection
with the x-axis? Sketch an example and give the equation.