The factor theorem is a direct result of the remainder theorem. We know that a polynomial function can be written in the form of f(x) = (x - k)q(x) + r. If f(k) = 0, then this tells us that x - k is a factor of f(x).

Test Objectives
• Demonstrate the ability to use the factor theorem to find factors
• Demonstrate the ability to use the factor theorem to factor a polynomial function
Factor Theorem Practice Test:

#1:

Instructions: Determine if g(x) is a factor of f(x).

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$$a)\hspace{.2em}f(x)=x^4 + 2x^3 + x + 2$$ $$g(x)=x + 1$$

$$b)\hspace{.2em}f(x)=x^4 - 2x^3 - 64x + 128$$ $$g(x)=x - 2$$

#2:

Instructions: Determine if g(x) is a factor of f(x).

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$$a)\hspace{.2em}f(x)=x^5 - 2x^4 + 5x^3 - 10x^2 + 6x - 12$$ $$g(x)=x + 3$$

$$b)\hspace{.2em}f(x)=x^5 - 5x^4 + 11x^3 - 55x^2 + 18x - 90$$ $$g(x)=x - 5$$

#3:

Instructions: Determine if g(x) is a factor of f(x).

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$$a)\hspace{.2em}f(x)=x^4 + 8x^3 + 13x^2 - 16x - 30$$ $$g(x)=x - 1$$

$$b)\hspace{.2em}f(x)=x^4 + 2x^3 + 27x + 54$$ $$g(x)=x + 3$$

#4:

Instructions: f(k) = 0, factor each.

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$$a)\hspace{.2em}f(x)=x^3 - 7x + 6$$ $$k=-3$$

$$b)\hspace{.2em}f(x)=x^3 - 2x^2 - x + 2$$ $$k=2$$

#5:

Instructions: f(k) = 0, factor each.

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$$a)\hspace{.2em}f(x)=x^3 - 21x - 20$$ $$k=5$$

$$b)\hspace{.2em}f(x)=x^3 - 7x^2 + 16x - 12$$ $$k=2$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}\text{Yes}$$

$$b)\hspace{.2em}\text{Yes}$$

#2:

Solutions:

$$a)\hspace{.2em}\text{No}$$

$$b)\hspace{.2em}\text{Yes}$$

#3:

Solutions:

$$a)\hspace{.2em}\text{No}$$

$$b)\hspace{.2em}\text{Yes}$$

#4:

Solutions:

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$$a)\hspace{.2em}f(x)=(x - 2)(x - 1)(x + 3)$$

$$b)\hspace{.2em}f(x)=(x - 1)(x + 1)(x - 2)$$

#5:

Solutions:

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$$a)\hspace{.2em}f(x)=(x + 1)(x + 4)(x - 5)$$

$$b)\hspace{.2em}f(x)=(x - 3)(x - 2)^2$$