Question 1

Given the polynomial $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is NOT a root of $$f(x)$$?

Accepted Answer

$$x = 4$$

Question 2

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, use synthetic division to determine the remainder when $$f(x) is $$divided by $$x - 2.$$

Accepted Answer

$$0$$

Question 3

Which of the following is the correct factorization of $$f(x) = x^3$ - 6x^2$ + 11x - 6$$?

Accepted Answer

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

Question 4

Suppose you use synthetic division to divide $$f(x) = x^3$ - 6x^2$ + 11x - 6 by$$ $$x - 1. $$What is the remainder?

Accepted Answer

$$0$$

Question 5

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ and you know $$x = 2 is a $$root, what is the quotient polynomial after dividing by $$x - 2$$?

Accepted Answer

$$x^2 - 4x + 3$

Question 6

Which theorem allows you to determine the remainder of $$f(x)$$ divided by $$x - a by $$evaluating $$f(a)$$?

Accepted Answer

The Remainder Theorem

Question 7

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ and you use synthetic division with $$x = 3$$, what is the remainder?

Accepted Answer

$$0$$

Question 8

After factoring $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following statements is true?

Accepted Answer

All roots are real and distinct.

Question 9

If you divide $$f(x) = x^3$ - 6x^2$ + 11x - 6 by$$ $$x - 1$$ using synthetic division, what is the resulting quadratic factor?

Accepted Answer

$$x^2 - 5x + 6$

Question 10

Which of the following is a correct application of the Factor Theorem to $$f(x) = x^3$ - 6x^2$ + 11x - 6$$?

Accepted Answer

If $$f(2) = 0$$, then $$x - 2 is a $$factor of $$f(x).$$

Question 11

A chemical reaction is modeled by $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, where $$x is $$the concentration of a reactant. What concentrations correspond to equilibrium points?

Accepted Answer

$$x = 1, 2, 3$$

Question 12

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ models the profit (in thousands of dollars) for a business, at which values of $$x$$ does the business break even?

Accepted Answer

$$x = 1, 2, 3$$

Question 13

Which of the following is the correct sequence of synthetic division steps for dividing $$f(x) = x^3$ - 6x^2$ + 11x - 6 by$$ $$x - 2$$?

Accepted Answer

Bring down 1, multiply by 2, add to -6, multiply by 2, add to 11, multiply by 2, add to -6.

Question 14

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ and you know $$x = 1 is a $$root, what is the next step to find all roots?

Accepted Answer

Divide $$f(x) by$$ $$x - 1$$ and factor the resulting quadratic.

Question 15

Which of the following is the correct application of the Rational Root Theorem to $$f(x) = x^3$ - 6x^2$ + 11x - 6$$?

Accepted Answer

Possible rational roots are $$\pm1, \pm2, \pm3, \pm6$$

Question 16

If you use synthetic division to divide $$f(x) = x^3$ - 6x^2$ + 11x - 6 by$$ $$x - 3$$, what is the resulting quadratic factor?

Accepted Answer

$$x^2 - 3x + 2$

Question 17

A medical dosage model is given by $$f(x) = x^3$ - 6x^2$ + 11x - 6. $$What dosages $$x$$ will result in zero net effect?

Accepted Answer

$$x = 1, 2, 3$$

Question 18

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$ and you divide by $$x - 1$$, what is the remainder and what does it mean?

Accepted Answer

Remainder is $$0$$, so $$x = 1 is a $$root.

Polynomial Functions and Roots: College Algebra Study Notes

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