Question 1

Which of the following best describes the purpose of synthetic division?

Accepted Answer

To divide a polynomial by a binomial of the form $$x - k$$ using only the coefficients.

Question 2

Given $$f(x) = 3x^3$ - 2x^2$ - 150$$, use synthetic division to find the quotient and remainder when divided by $$x + 4.$$

Accepted Answer

Quotient: $$3x^2 + 10x + 40$, Remainder: 10$

Question 3

What is the Remainder Theorem?

Accepted Answer

If a polynomial $$f(x) is $$divided by $$x - k$$, the remainder is $$f(k).$$

Question 4

A chemist is modeling the concentration of a substance with $$f(x) = 5x^3$ - 6x^2$ - 28x - 2. $$What is $$f(-2)$$?

Accepted Answer

$$-10$$

Question 5

Which of the following is a necessary step when setting up synthetic division if a term is missing in the polynomial?

Accepted Answer

Insert $$0 as $$the coefficient for the missing term.

Question 6

If $$f(x) = x^3$ - 2x^2$ + 4x + 2$$ and you use synthetic division with $$k = 1 + 2i$$, what is the remainder?

Accepted Answer

$$0$$

Question 7

A company’s profit is modeled by $$f(x) = x^4$ + 3x^2$ - 4. $$Use the Remainder Theorem to determine if $$x = 1 is a $$zero of $$f(x).$$

Accepted Answer

Yes, because $$f(1) = 0 is $$true.

Question 8

Which of the following statements is true about the zeros of a polynomial function?

Accepted Answer

A zero $$k of$$ $$f(x)$$ satisfies $$f(k) = 0.$$

Question 9

A medical researcher uses $$f(x) = x^3$ + 2x^2$ - x - 2 to $$model a process. Use synthetic division to test if $$x = 1 is a $$zero.

Accepted Answer

Yes, because the remainder is $$0.$$

Question 10

Which of the following is the correct setup for synthetic division of $$f(x) = 2x^3$ - 5x + 7 by$$ $$x - 1$$?

Accepted Answer

Coefficients: $$2, 0, -5, 7$$; $$k = 1$$

Question 11

If the remainder from synthetic division is not zero, what does this indicate about the proposed zero $$k$$?

Accepted Answer

$$k is $$not a zero of $$f(x).$$

Question 12

A polynomial $$f(x) is $$divided by $$x - 3$$ and the remainder is $$5. $$What is $$f(3)$$?

Accepted Answer

$$5$$

Question 13

Which of the following is a real-world application of the Remainder Theorem?

Accepted Answer

Quickly checking if a proposed solution to a chemical reaction model is valid.

Question 14

If $$f(x) = x^3$ + 2x^2$ - x - 2$$, what is the result of dividing $$f(x) by$$ $$x - 1$$ using synthetic division?

Accepted Answer

Quotient: $$x^2 + 3x + 2$, Remainder: 0$

Question 15

A student divides $$f(x) = 4x^3$ - 8x + 5 by$$ $$x + 2$$ using synthetic division. What value should be used for $$k$$?

Accepted Answer

$$-2$$

Question 16

Which of the following is NOT a step in synthetic division?

Accepted Answer

Divide each coefficient by the divisor.

Question 17

A polynomial $$f(x) is $$divided by $$x - k$$ and the remainder is $$0. $$What does this tell you about $$k$$?

Accepted Answer

$$k is a $$zero of $$f(x).$$

Question 18

If $$f(x) = 2x^3$ + 3x^2$ - x + 4$$ and you want to find $$f(-1)$$ using synthetic division, what is the remainder?

Accepted Answer

Question 19

A business uses $$f(x) = x^3$ - 6x^2$ + 11x - 6 to $$model profit. Which value of $$x is $$NOT a zero of $$f(x)$$?

Accepted Answer

$$x = 4$$

3-02: Study Notes: Synthetic Division, Remainder Theorem, and Potential Zeros of Polynomial Functions

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