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

Study Guide - Practice Questions

Study Guide - Flashcards