Question 1

Which of the following statements best describes the Rational Zero Theorem?

Accepted Answer

It provides a list of all possible rational zeros of a polynomial function with integer coefficients.

Question 2

Given the polynomial $$f(x) = 2x^3$ - 3x^2$ - 8x + 12$$, which of the following is NOT a possible rational zero according to the Rational Zero Theorem?

Accepted Answer

$$6$$

Question 3

If $$f(x) = x^3$ - 4x^2$ + x + 6$$, which of the following is a zero of $$f(x)$$?

Accepted Answer

$$2$$

Question 4

A chemist models the concentration of a substance with $$C(t) = t^3$ - 6t^2$ + 11t - 6. $$What is the concentration when $$t is a $$zero of $$C(t)$$?

Accepted Answer

The concentration is zero.

Question 5

Which of the following is true about the roots of the polynomial $$x^4$$ + $$2x^2 + 5 = 0$?

Accepted Answer

All roots are complex and occur in conjugate pairs.

Question 6

If $$2 + 3i is a $$root of a polynomial with real coefficients, which of the following must also be a root?

Accepted Answer

$$2 - 3i$$

Question 7

A polynomial equation of degree 5 has how many roots, counting multiplicities?

Accepted Answer

5

Question 8

Which of the following is the correct factorization of $$x^3$$ - $$6x^2 + 11x - 6$ given that its zeros are 1$, 2$, and 3$?

Accepted Answer

$$(x - 1)(x - 2)(x - 3)$$

Question 9

Using Descartes’s Rule of Signs, how many positive real zeros does $$f(x) = x^3$ - 2x^2$ + 3x - 4$$ possibly have?

Accepted Answer

3 or 1

Question 10

A pharmaceutical company wants to design a drug release modeled by $$f(x) = x^4$ - 5x^2$ + 4. If$$ $$x$$ represents time, what does a zero of $$f(x)$$ represent in this context?

Accepted Answer

The time when the drug concentration is zero.

Question 11

Given the polynomial $$f(x) = x^3$ + 2x^2$ - x - 2$$, which of the following is a rational zero?

Accepted Answer

$$-2$$

Question 12

If a polynomial $$f(x)$$ has a repeated root at $$x = 1$$, which of the following must be true?

Accepted Answer

$$f(1) = 0$$ and $$f'(1) = 0$$

Question 13

Which of the following is a correct application of the Linear Factorization Theorem?

Accepted Answer

Every polynomial of degree $$n$$ can be written as a product of $$n$$ linear factors over the complex numbers.

Question 14

If $$f(x) = x^4$ - 2x^2$ + 1$$, what are the zeros of $$f(x)$$?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 15

A polynomial $$f(x)$$ has the following sign changes in $$f(x)$$: $$+$$, $$-$$, $$+$$, $$-$$, $$+. $$According to Descartes’s Rule of Signs, what is the maximum number of positive real zeros?

Accepted Answer

4

Zeros of Polynomial Functions – College Algebra Study Notes

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