$$f(3) = 3^2$ - 5 \times 3 + 6 = 9 - 15 + 6 = 0$$

$$f(2) = 2^2$ - 4 \times 2 + 4 = 4 - 8 + 4 = 0$$

Question 1

Which of the following is the correct setup for synthetic division when dividing $$3x^3$$ - $$7x^2 + 5x - 2$ by x - 4$?

Accepted Answer

Use 4 as the divisor and the coefficients 3, -7, 5, -2.

Question 2

What is the remainder when $$2x^3$$ - $$3x^2 + 0x - 18$ is divided by x - 2$ using synthetic division?

Accepted Answer

$$-14$$

Question 3

According to the Remainder Theorem, what is the remainder when $$f(x) = x^3$ - 4x^2$ + 9x - 6 is $$divided by $$x - 2$$?

Accepted Answer

$$10$$

Question 4

If the remainder from synthetic division of $$f(x) by$$ $$x - k is $$zero, what does this imply about $$k$$?

Accepted Answer

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

Question 5

Use synthetic division to determine if $$x = 1 is a $$zero of $$f(x) = x^3$ + 2x^2$ + 2x - 4.$$

Accepted Answer

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

Question 6

Which of the following best describes the Remainder Theorem?

Accepted Answer

The remainder when $$f(x) is $$divided by $$x - k is$$ $$f(k).$$

Question 7

A chemical reaction rate is modeled by $$f(x) = 2x^3$ - 5x^2$ + 4x - 6. $$What is the value of $$f(3)$$ using synthetic division?

Accepted Answer

$$15$$

Question 8

When dividing $$5x^3$$ - $$6x^2 - 28x - 2$ by x + 2$, what is the remainder?

Accepted Answer

$$-10$$

Question 9

A polynomial $$f(x) is $$divided by $$x - k$$ and the remainder is $$r. $$Which equation is always true?

Accepted Answer

$$f(x) = (x - k)q(x) + r$$

Question 10

A medical dosage model is given by $$f(x) = x^2$ - 5x + 6. $$What is the remainder when divided by $$x - 3$$?

Accepted Answer

$$f(3) = 3^2$ - 5 	imes 3 + 6 = 9 - 15 + 6 = 0$$

Question 11

Which step is performed first in synthetic division?

Accepted Answer

Bring down the leading coefficient.

Question 12

If $$f(x) = x^3$ - 3x^2$ + 0x - 6$$, what is the remainder when divided by $$x - 1$$?

Accepted Answer

$$-8$$

Question 13

A polynomial $$f(x) is $$divided by $$x - k$$ and the remainder is $$0. $$What does this mean in the context of finding roots?

Accepted Answer

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

Question 14

Which of the following is a correct interpretation of the synthetic division diagram for dividing $$ax^3$$ + $$bx^2 + cx + d$ by x - k$?

Accepted Answer

The bottom row gives the coefficients of the quotient and the remainder.

Question 15

A business profit model is $$f(x) = x^2$ - 4x + 4. $$What is the remainder when divided by $$x - 2$$?

Accepted Answer

$$f(2) = 2^2$ - 4 	imes 2 + 4 = 4 - 8 + 4 = 0$$

Question 16

When using synthetic division, what does each diagonal multiplication represent?

Accepted Answer

Multiplying the previous sum by the divisor value $$k.$$

Question 17

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

Accepted Answer

$$-10$$

Question 18

A polynomial $$f(x) is $$divided by $$x - k$$ and the remainder is $$r. $$Which of the following is true?

Accepted Answer

$$f(k) = r$$

Question 19

Which of the following is NOT a valid use of synthetic division?

Accepted Answer

Dividing by $$x^2 - k$

Synthetic Division, Remainder Theorem, and Zeros of Polynomial Functions

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