The remainder theorem gives us a quicker method to evaluate a polynomial for a given value. When we work with a polynomial, we can write the polynomial as: f(x) = (x - k)q(x) + r, where (x - k) is our divisor, q(x) is the quotient, and r is the remainder. If we want to find f(k), we see that f(k) = r. So f(k) is just equal to the remainder from dividing f(x) by x - k.

Test Objectives
• Demonstrate the ability to evaluate a polynomial function using the remainder theorem
• Demonstrate the ability to determine if a given value is a zero
Remainder Theorem Practice Test:

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#1:

Instructions: Find f(k).

$$a)\hspace{.2em}f(x)=4x^6 - 2x^5 - 6x^3 - x^2 + 2x + 1$$ $$k=-1$$

$$b)\hspace{.2em}f(x)=x^6 + 5x^5 + 4x^4 - 4x^3 + 6x^2 + x - 4$$ $$k=-3$$

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#2:

Instructions: Find f(k).

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

$$b)\hspace{.2em}f(x)=2x^4 - 3x^3 - 6x^2 - 1$$ $$k=2$$

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#3:

Instructions: Find f(k).

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

$$b)\hspace{.2em}f(x)=x^2 - x - 4$$ $$k=3 - i$$

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#4:

Instructions: Determine if k is a zero.

$$a)\hspace{.2em}f(x)=x^4 + 6x^3 + 12x^2 + 8x - 1$$ $$k=-1$$

$$b)\hspace{.2em}f(x)=x^5 + 7x^4 + 18x^3 + 20x^2 + 8x$$ $$k=-1$$

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#5:

Instructions: Determine if k is a zero.

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

$$b)\hspace{.2em}f(x)=x^4 - 2x^3 + 4x^2 + 2x - 5$$ $$k=1 + 2i$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}f(-1)=10$$

$$b)\hspace{.2em}f(-3)=-7$$

#2:

Solutions:

$$a)\hspace{.2em}f(3)=-25$$

$$b)\hspace{.2em}f(2)=-17$$

#3:

Solutions:

$$a)\hspace{.2em}f(2 + i)=-6 - i$$

$$b)\hspace{.2em}f(3 - i)=1 - 5i$$

#4:

Solutions:

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

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

#5:

Solutions:

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

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