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

$$4$$

Question 3

If $$f(x) = x^4$ - 5x^2$ + 6$$, how many real zeros does $$f(x)$$ have?

Accepted Answer

2

Question 4

Which of the following is a property of the roots of a polynomial equation with real coefficients?

Accepted Answer

Complex roots always occur in conjugate pairs.

Question 5

A chemist models the concentration of a substance with $$C(t) = t^3$ - 4t^2$ + t + 6. $$Using the Rational Zero Theorem, which is a possible rational time (in hours) when the concentration is zero?

Accepted Answer

$$-3$$

Question 6

If $$f(x) = x^3$ + 2x^2$ - x - 2$$, and $$x = 1 is a $$zero, what is the depressed quadratic factor after dividing by $$(x-1)$$?

Accepted Answer

$$x^2 + 3x + 2$

Question 7

Which theorem guarantees that a polynomial of degree $$n$$ has exactly $$n$$ roots (counting multiplicities) in the complex number system?

Accepted Answer

The Fundamental Theorem of Algebra

Question 8

If $$f(x) = x^3$ - 2x^2$ + 4x - 8$$ and $$x = 2 is a $$zero, what is the complete factorization of $$f(x)$$ over the real numbers?

Accepted Answer

$$(x-2)(x^2 + 4$$)$$

Question 9

A polynomial $$f(x)$$ has real coefficients and a root at $$x = 2 + 3i. $$What other root must $$f(x)$$ have?

Accepted Answer

$$x = 2 - 3i$$

Question 10

Using Descartes’s Rule of Signs, how many positive real zeros does $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ have?

Accepted Answer

3 or 1

Question 11

Which of the following is the correct application of the Linear Factorization Theorem for $$f(x) = x^2$ - 5x + 6$$?

Accepted Answer

$$f(x) = (x-2)(x-3)$$

Question 12

A medical researcher models the spread of a virus with $$V(t) = t^4$ - 4t^2$. How many distinct real zeros does V(t$$)$$ have?

Accepted Answer

4

Question 13

If $$f(x) = x^3$ - 7x + 6$$, which of the following is a rational zero?

Accepted Answer

$$x = 2$$

Question 14

Which of the following best describes Descartes’s Rule of Signs?

Accepted Answer

It provides the possible number of positive and negative real zeros based on sign changes.

Question 15

Given $$f(x) = x^3$ + x^2$ - 4x - 4$$, use synthetic division to test if $$x = -1 is a $$zero.

Accepted Answer

Yes, because the remainder is 0.

Question 16

If a polynomial equation has degree 5, what is the maximum number of positive real zeros it can have according to Descartes’s Rule of Signs?

Accepted Answer

5

Zeros of Polynomial Functions – College Algebra Study Notes

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