7.14: The Factor Theorem
A rectangular shaped container is built in such a way that its volume can be represented by the polynomial
a) Factor the polynomial.
b) If
Watch This
The Factor Theorem and the Remainder Theorem
Guidance
You know techniques for factoring quadratics and special cases of cubics, but what about other cubics or higher degree polynomials? With the factor theorem, you can attempt to factor these types of polynomials. The factor theorem states that if
 Guess factors of the given polynomial
p(x) . Factors should be of the form(x−a) wherea is a factor of the constant term of the polynomial divided by a factor of the first coefficient of the polynomial.  Test potential factors by checking
p(a) . Ifp(a)=0 , thenx−a is a factor of the polynomial.  Divide the polynomial by one of its factors.
 Repeat Steps 2 and 3 until the result is a quadratic expression that you can factor using other methods.
Example A
Use the factor theorem to determine if
Solution: If
Now that you have one of the factors, use division to find the others.
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
So:
Example B
Use the factor theorem to determine if
Solution: If
Now that you have one of the factors, use division to find the other factor.
Step 1: Divide the first term in the numerator by the first term in the denominator; put this in your answer. Therefore
Step 2: Multiply the denominator by this number (variable) and put it below your numerator, subtract and get your new polynomial.
Step 3: Repeat the process until you cannot repeat it anymore.
Therefore:
So:
Example C
Factor
Solution: In order to begin to find the factors, look at the number –10 and find the factors of this number. The factors of –10 are –1, 1, –2, 2, –5, 5, –10, 10. Next, start testing the factors to see if you get a remainder of zero.
Now that you have one of the factors, use division to find the others.
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
So :
Therefore:
Concept Problem Revisited
A rectangular shaped container is built in such a way that its volume can be represented by the polynomial
a) In order to begin to find the factors, look at the number 12 and find the factors of this number. The factors of 12 are
Now that you have one of the factors, use division to find the others.
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
So:
b) If
Therefore. the dimensions of the container are
Vocabulary
 The Factor theorem

The factor theorem states that if
p(a)=0 , thenx−a is a factor ofp(x) .
Guided Practice
1. Determine if
2. Factor:
3. A tennis court is being built where the volume is represented by the polynomial
Answers:
1. \begin{align*}e=3:2(3)^3(3)^2+(3)1=67\end{align*}
2. In order to begin to find the factors, look at the number –6 and find the factors of this number. The factors of –6 are \begin{align*}\pm 1, \pm 2, \pm 3,\end{align*}

\begin{align*}& x^3+4x^2+x6\\
& x=1:(1)^3+4(1)^2+(1)6=4 \quad (\text{NOT a factor})\\
& x=1:(1)^3+4(1)^2+(1)6=0 \quad (\text{IS a factor})\end{align*}
x3+4x2+x−6x=−1:(−1)3+4(−1)2+(−1)−6=−4(NOT a factor)x=1:(1)3+4(1)2+(1)−6=0(IS a factor)  Now that you have one of the factors, use division to find the others.
 Step 1:
 Step 2:
 Step 3:
 Step 4:
 Step 5:
 Step 6:

Therefore: \begin{align*}x^3+4x^2+x6=(x1)(x^2+5x+6)=(x1)(x+2)(x+3)\end{align*}
x3+4x2+x−6=(x−1)(x2+5x+6)=(x−1)(x+2)(x+3)
3. Start by testing the factor \begin{align*}L + 1\end{align*}

\begin{align*}p(L)&=3L^3+8L^2+3L2L=1:\\
p(L)&=3(1)^3+8(1)^2+3(1)2\\
p(1)&=0 \quad (\text{IS a factor})\end{align*}
p(L)p(L)p(1)=3L3+8L2+3L−2L=−1:=3(−1)3+8(−1)2+3(−1)−2=0(IS a factor)  Now that you have one of the factors, use division to find the others.
 Step 1:
 Step 1:
 Step 2:
 Step 3:
 Step 4:
 Step 5:
 So:

\begin{align*}p(L)&=3L^3+8L^2+3L2\\
p(L)&=(L+1)(3L^2+5L2)\\
p(L)&=(L+1)(3L1)(L+2)\end{align*}
p(L)p(L)p(L)=3L3+8L2+3L−2=(L+1)(3L2+5L−2)=(L+1)(3L−1)(L+2)  If \begin{align*}L = 5 \ ft\end{align*}, what are the dimensions of the container?

\begin{align*}(L+1)&=5+1=6\\
(3L1)&=3(5)1=14\\
(L+2)&=5+2=7\end{align*}
 Therefore the dimensions of the container are \begin{align*}6 \ ft \times 14 \ ft \times 7 \ ft\end{align*}.
Practice
Determine if \begin{align*}a  4\end{align*} is a factor of each of the following.
 \begin{align*}a^35a^2+3a+4\end{align*}
 \begin{align*}3a^27a20\end{align*}
 \begin{align*}a^4+3a^3+5a^216\end{align*}
 \begin{align*}a^42a^38a^2+3a4\end{align*}
 \begin{align*}2a^45a^37a^221a+4\end{align*}
Factor each of the following:
 \begin{align*}x^3+2x^2+2x+1\end{align*}
 \begin{align*}x^3+x^2x1\end{align*}
 \begin{align*}2x^35x^2+2x+1\end{align*}
 \begin{align*}2b^3+4b^23b6\end{align*}
 \begin{align*}3c^34c^2c+2\end{align*}
 \begin{align*}2x^313x^2+17x+12\end{align*}
 \begin{align*}x^3+2x^2x2\end{align*}
 \begin{align*}3x^3+2x^253x+60\end{align*}
 \begin{align*}x^37x^2+7x+15\end{align*}
 \begin{align*}x^4+4x^37x^234x24\end{align*}
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Description
Learning Objectives
Here you will learn how to factor a polynomial using the factor theorem.