Question 1

Suppose you divide $$f(x) = x^3$ + 3x^2$ + 2x - 1 by$$ $$x + 2. $$What is the remainder?

Accepted Answer

$$-1$$

Question 2

According to the Factor Theorem, which of the following statements is true?

Accepted Answer

If $$f(c) = 0$$, then $$x - c is a $$factor of $$f(x).$$

Question 3

Given $$f(x) = x^3$ - 8x^2$ - 7x - 12$$, which of the following is a possible rational zero according to the Rational Zeros Theorem?

Accepted Answer

$$\frac{12}{1}$$

Question 4

Use Descartes' Rule of Signs to determine the maximum number of positive real zeros for $$f(x) = x^4$ + 2x^2$ - 5x - 6.$$

Accepted Answer

$$2$$

Question 5

If $$f(x) = x^4$ - x^2$ + 2x^2$ - 4x - 8$$, what is the sum of the coefficients of $$f(x)$$?

Accepted Answer

$$-9$$

Question 6

Which of the following statements about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f(a)$$ and $$f(b)$$ are of opposite sign, then $$f(x)$$ has at least one real zero between $$a$$ and $$b.$$

Question 7

A chemist is modeling the concentration of a reactant with $$f(x) = x^3$ - 6x^2$ + 11x - 6. $$According to the Rational Zeros Theorem, which of the following is NOT a possible rational zero?

Accepted Answer

$$4$$

Question 8

If $$f(x) = x^3$ - 4x^2$ + x - 4 is $$divided by $$x - 1$$, what is the remainder?

Accepted Answer

Question 9

A medical researcher uses $$f(x) = x^4$ - 2x^3$ - 4x^2$ + x - 1 to $$model the spread of a virus. Using Descartes' Rule of Signs, what is the maximum number of negative real zeros?

Accepted Answer

$$2$$

Question 10

Which of the following is a correct step in approximating the real zeros of a polynomial function?

Accepted Answer

Divide the interval $$[a, a+1]$$ into 10 subintervals and check the sign of $$f(x) at $$each endpoint.

Question 11

If $$f(x) = x^3$ - 2x^2$ + 3x - 4$$ and $$g(x) = x - 2$$, what is the remainder when $$f(x) is $$divided by $$g(x)$$?

Accepted Answer

$$0$$

Question 12

A polynomial $$f(x) of $$degree 5 with real coefficients must have at least how many real zeros?

Accepted Answer

$$1$$

Question 13

Given $$f(x) = x^4$ - 2x^3$ - 4x^2$ + x - 1$$, which of the following is a possible upper bound for the real zeros?

Accepted Answer

$$3$$

Question 14

If $$f(x) = x^3$ + 2x^2$ - 5x - 6$$, which of the following is a possible rational zero?

Accepted Answer

$$\frac{6}{1}$$

Question 15

A polynomial $$f(x)$$ has integer coefficients and $$f(2) = 0. $$Which of the following must be true?

Accepted Answer

$$x - 2 is a $$factor of $$f(x).$$

Question 16

If $$f(x) = x^3$ - 4x^2$ + x - 4$$ and $$g(x) = x - 1$$, what is $$f(1)$$?

Accepted Answer

$$-6$$

Question 17

Which theorem guarantees that a continuous polynomial function that changes sign between $$a$$ and $$b$$ must have a real zero between $$a$$ and $$b$$?

Accepted Answer

Intermediate Value Theorem

Question 18

A polynomial $$f(x)$$ has a leading coefficient of $$1$$ and degree $$4. $$Which of the following statements is true about the number of real zeros?

Accepted Answer

It can have $$0$$, $$2$$, or $$4$$ real zeros.

Question 19

If $$f(x) = x^3$ - 2x^2$ + 3x - 4$$ and $$f(2) = 0$$, what does this imply?

Accepted Answer

$$x - 2 is a $$factor of $$f(x).$$

The Real Zeros of a Polynomial Function: Key Theorems and Methods

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