Factor Theorem Worksheets (printable, online, answers, examples) - Free Printable

To determine whether each binomial is a factor of the given polynomial, we use the Factor Theorem. The Factor Theorem states that a polynomial \( P(x) \) has a factor \( (x - c) \) if and only if \( P(c) = 0 \). This means we need to evaluate the polynomial at the value of \( x \) that makes the binomial zero.

Let's solve each problem step by step.

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1. Determine whether \( (x - 3) \) is a factor of \( 9x^3 - 18x^2 - 2x + 4 \)

#### Step 1: Identify \( c \)
The binomial is \( x - 3 \), so \( c = 3 \).

#### Step 2: Define the polynomial
\[ P(x) = 9x^3 - 18x^2 - 2x + 4 \]

#### Step 3: Evaluate \( P(3) \)
\[ P(3) = 9(3)^3 - 18(3)^2 - 2(3) + 4 \]
\[ P(3) = 9(27) - 18(9) - 2(3) + 4 \]
\[ P(3) = 243 - 162 - 6 + 4 \]
\[ P(3) = 79 \]

#### Step 4: Check if \( P(3) = 0 \)
Since \( P(3) = 79 \neq 0 \), \( (x - 3) \) is not a factor of \( 9x^3 - 18x^2 - 2x + 4 \).

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2. Determine whether \( (x + 4) \) is a factor of \( x^3 + 4x^2 + 4x + 16 \)

#### Step 1: Identify \( c \)
The binomial is \( x + 4 \), so \( c = -4 \).

#### Step 2: Define the polynomial
\[ P(x) = x^3 + 4x^2 + 4x + 16 \]

#### Step 3: Evaluate \( P(-4) \)
\[ P(-4) = (-4)^3 + 4(-4)^2 + 4(-4) + 16 \]
\[ P(-4) = -64 + 4(16) - 16 + 16 \]
\[ P(-4) = -64 + 64 - 16 + 16 \]
\[ P(-4) = 0 \]

#### Step 4: Check if \( P(-4) = 0 \)
Since \( P(-4) = 0 \), \( (x + 4) \) is a factor of \( x^3 + 4x^2 + 4x + 16 \).

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3. Determine whether \( (x - 4) \) is a factor of \( 4x^3 - 16x^2 + 25x - 100 \)

#### Step 1: Identify \( c \)
The binomial is \( x - 4 \), so \( c = 4 \).

#### Step 2: Define the polynomial
\[ P(x) = 4x^3 - 16x^2 + 25x - 100 \]

#### Step 3: Evaluate \( P(4) \)
\[ P(4) = 4(4)^3 - 16(4)^2 + 25(4) - 100 \]
\[ P(4) = 4(64) - 16(16) + 25(4) - 100 \]
\[ P(4) = 256 - 256 + 100 - 100 \]
\[ P(4) = 0 \]

#### Step 4: Check if \( P(4) = 0 \)
Since \( P(4) = 0 \), \( (x - 4) \) is a factor of \( 4x^3 - 16x^2 + 25x - 100 \).

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4. Determine whether \( (x - 3) \) is a factor of \( x^3 - 2x^2 + 3x - 6 \)

#### Step 1: Identify \( c \)
The binomial is \( x - 3 \), so \( c = 3 \).

#### Step 2: Define the polynomial
\[ P(x) = x^3 - 2x^2 + 3x - 6 \]

#### Step 3: Evaluate \( P(3) \)
\[ P(3) = (3)^3 - 2(3)^2 + 3(3) - 6 \]
\[ P(3) = 27 - 2(9) + 9 - 6 \]
\[ P(3) = 27 - 18 + 9 - 6 \]
\[ P(3) = 12 \]

#### Step 4: Check if \( P(3) = 0 \)
Since \( P(3) = 12 \neq 0 \), \( (x - 3) \) is not a factor of \( x^3 - 2x^2 + 3x - 6 \).

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5. Determine whether \( (x - 5) \) is a factor of \( x^3 - 5x^2 + 4x - 20 \)

#### Step 1: Identify \( c \)
The binomial is \( x - 5 \), so \( c = 5 \).

#### Step 2: Define the polynomial
\[ P(x) = x^3 - 5x^2 + 4x - 20 \]

#### Step 3: Evaluate \( P(5) \)
\[ P(5) = (5)^3 - 5(5)^2 + 4(5) - 20 \]
\[ P(5) = 125 - 5(25) + 20 - 20 \]
\[ P(5) = 125 - 125 + 20 - 20 \]
\[ P(5) = 0 \]

#### Step 4: Check if \( P(5) = 0 \)
Since \( P(5) = 0 \), \( (x - 5) \) is a factor of \( x^3 - 5x^2 + 4x - 20 \).

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Final Answers

\[
\boxed{
\begin{aligned}
1. & \quad \text{(x - 3) is not a factor of } 9x^3 - 18x^2 - 2x + 4. \\
2. & \quad \text{(x + 4) is a factor of } x^3 + 4x^2 + 4x + 16. \\
3. & \quad \text{(x - 4) is a factor of } 4x^3 - 16x^2 + 25x - 100. \\
4. & \quad \text{(x - 3) is not a factor of } x^3 - 2x^2 + 3x - 6. \\
5. & \quad \text{(x - 5) is a factor of } x^3 - 5x^2 + 4x - 20.
\end{aligned}
}
\]