Perform the polynomial division: $(x^2 + 5x + 7) \div (x + 2)$. What is the result?
$x - 3 + \frac{1}{x+2}$
$x + 3$
$x + 3 + \frac{1}{x+2}$
$x + 7 + \frac{21}{x+2}$
When $x^3 - 4x^2 - 7x + 10$ is divided by $x - 5$, what is the quotient?
$x^2 - 9x + 38$
$x^2 + x - 2$
$x^2 + x + 2$
$x^2 - x - 2$
Find the remainder when $2x^3 - 5x^2 + 3x - 1$ is divided by $x - 3$.
$23$
$-1$
$0$
$17$
A polynomial has a missing term. Divide $4x^3 - 9x + 5$ by $2x - 1$. What is the result? Remember to write the dividend as $4x^3 + 0x^2 - 9x + 5$.
$2x^2 + x - 4$
$2x^2 - x - 5$
$2x^2 - 4$
$2x^2 + x - 4 + \frac{1}{2x-1}$
If the result of dividing a polynomial $P(x)$ by $(x+4)$ is $x^2 - 2x + 3$ with a remainder of $-5$, what is the polynomial $P(x)$?
$x^3 + 6x^2 + 11x + 7$
$x^3 - 2x^2 + 3x - 5$
$x^3 + 2x^2 - 5x + 7$
$x^3 + 2x^2 - 5x - 17$
Use synthetic division to find the quotient of $(x^3 + 2x^2 - 5x - 6) \div (x - 2)$.
$x^2 - 4x + 3$
$x^3 + 4x^2 + 3x$
$x^2 + 4x + 3$
$x^2 + 3$
What is the remainder when the polynomial $P(x) = 2x^3 - 3x^2 + 4x - 7$ is divided by $x + 1$?
$-16$
$-6$
$16$
When performing synthetic division for $(x^4 - 3x^2 + 5x - 1) \div (x - 3)$, what is the correct quotient?
$x^3 + 3x^2 + 6x + 23$
$x^2 + 5$
$x^3 + 6x + 23$
$x^3 - 3x^2 + 6x - 13$
Using synthetic division, find the remainder of $(4x^3 - 5x + 2) \div (x + 2)$.
$-20$
$24$
$28$
For which of the following division problems can synthetic division be directly applied in its standard form?
$(x^3 + 1) \div (x^2 + 1)$
$(x^5 - 3x^2 + 2) \div (x + 5)$
$(x^2 + x + 1) \div (x^3 + 2)$
$(2x^4 - x) \div (2x - 1)$
According to the Remainder Theorem, what is the remainder when the polynomial $P(x) = x^3 + 2x^2 - 5x + 1$ is divided by $(x - 2)$?
$7$
$5$
$11$
Find the remainder when $P(x) = 2x^3 - x^2 + 3x - 4$ is divided by $(x + 1)$.
$-10$
$-8$
When the polynomial $f(x) = x^3 + kx^2 - 3x + 5$ is divided by $(x - 1)$, the remainder is $7$. What is the value of $k$?
$k = 0$
$k = 9$
$k = -10$
$k = 4$
What is the remainder when the polynomial $P(x) = 4x^3 - 5x + 2$ is divided by the linear factor $(2x - 1)$?
$4$
$1$
The Remainder Theorem states that if a polynomial $P(x)$ is divided by a linear factor $(x - a)$, the remainder is equal to which of the following?
$P(-a)$
$P(0)$
$a$
$P(a)$
According to the Factor Theorem, which of the following is a factor of the polynomial $P(x) = x^3 - 2x^2 + 5x - 10$?
$(x-2)$
$(x+2)$
$(x+1)$
$(x-1)$
If $(x+3)$ is a factor of the polynomial $f(x) = x^3 + kx^2 - x - 21$, what is the value of the constant $k$?
$k = -1$
$k = 1$
$k = -3$
$k = 3$
The Factor Theorem states that for a polynomial $P(x)$, a binomial $(x-a)$ is a factor if and only if:
$P(x) = a$
$P(a) = 0$
$P(0) = a$
The remainder is $a$
Given the polynomial $g(x) = 2x^3 - 5x^2 - 4x + 3$, which of the following is a factor?
If $(2x-1)$ is a factor of a polynomial $P(x)$, which of the following statements must be true based on the Factor Theorem?
$P(1/2) = 0$
$P(-1/2) = 0$
$P(1) = 0$
$P(-1) = 0$
Completely factor the polynomial $P(x) = x^3 - 2x^2 + 5x - 10$.
$(x+2)(x^2-5)$
$(x+5)(x^2-2)$
$(x-2)(x^2-5)$
$(x-2)(x^2+5)$
Which of the following is a factor of the polynomial $f(x) = x^3 - 7x - 6$?
$(x+3)$
Factor the polynomial $g(x) = x^4 - 81$ completely.
$(x-3)(x+3)(x^2+9)$
$(x^2-9)(x^2+9)$
$(x-3)^2(x+3)^2$
$(x-3)(x+3)(x-3i)(x+3i)$
What is the complete factorization of the polynomial $h(t) = t^3 + 64$?
$(t+8)(t-8)$
$(t+4)(t^2+4t+16)$
$(t-4)(t^2+4t+16)$
$(t+4)(t^2-4t+16)$
Factor the polynomial $P(y) = y^4 - 3y^2 - 4$ completely.
$(y^2-1)(y^2+4)$
$(y-2)(y+2)(y^2+1)$
$(y-2)(y+2)(y^2-1)$
$(y-1)(y+1)(y^2-4)$
According to the Rational Root Theorem, which of the following is a complete list of all possible rational roots of the polynomial $P(x) = x^3 - 4x^2 + x + 6$?
$1, 2, 3, 6$
$\pm1, \pm2, \pm3$
$\pm1, \pm6$
$\pm1, \pm2, \pm3, \pm6$
Consider the polynomial $f(x) = 3x^3 + 2x^2 - 7x + 2$. Which of the following is a list of all possible rational roots?
$\pm1, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}$
$\pm1, \pm2$
$\pm1, \pm2, \pm3, \pm\frac{1}{2}, \pm\frac{3}{2}$
$\pm1, \pm2, \pm\frac{1}{3}, \pm\frac{2}{3}$
For the polynomial $P(x) = 2x^4 - 5x^3 + x^2 - 4$, which of the following numbers cannot be a rational root?
$2/3$
$2$
$-1/2$
What is the primary purpose of the Rational Root Theorem?
To provide a finite list of possible rational roots to test.
To find all irrational and complex roots of a polynomial.
To determine the exact number of positive real roots a polynomial has.
To guarantee that every polynomial has at least one rational root.
A polynomial is given by $f(x) = 6x^3 - 11x^2 - 4x + 4$. Using the Rational Root Theorem, which of the following values is a possible rational root of $f(x)$?
$1/4$
$3$
$3/4$
A 'zero' of a polynomial function $P(x)$ is a value of $x$ for which...
the graph of $P(x)$ crosses the y-axis
$P(x) > 0$
$P(0)$ is defined
$P(x) = 0$
Find all the zeros of the polynomial function $f(x) = (x-2)(x+3)(x-5)$.
${-2, -3, -5}$
${2, 3, 5}$
${2, -3, 5}$
${-2, 3, -5}$
Which of the following is a zero of the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$?
$x = 2$
$x = -1$
$x = -3$
$x = 3$
What are all the real zeros of the polynomial function $g(x) = x^3 - x^2 + 4x - 4$?
${1, -4}$
${1, 2, -2}$
${1}$
${-1, 4}$
Consider the polynomial function $h(x) = x(x+1)^3(x-4)^2$. What is the multiplicity of the zero at $x = -1$?
$6$
What are the solutions to the polynomial equation $x^3 - 2x^2 - 5x + 6 = 0$?
${1, 2, 3}$
${-1, 2, -3}$
${-1, -2, -3}$
${1, -2, 3}$
Find the solution set for the inequality $x^2 - 3x - 10 > 0$.
$(-\infty, -2) \cup (5, \infty)$
$(-\infty, -2] \cup [5, \infty)$
$[-2, 5]$
$(-2, 5)$
Which of the following is the solution to the inequality $(x+1)(x-3)(x-5) \le 0$?
$[-1, 3] \cup (5, \infty)$
$(-\infty, -1] \cup [3, 5]$
$[-1, 3] \cup [5, \infty)$
$(-\infty, -1) \cup (3, 5)$
Solve the polynomial equation $x^3 + 2x^2 - 9x - 18 = 0$ by factoring.
${2, 9}$
${-2, 3, -3}$
${-2, 9}$
${2, 3, -3}$
What is the solution set for the polynomial inequality $(x-4)^2(x+2) > 0$?
$(-\infty, -2)$
$(-2, 4)$
$(-2, \infty)$
$(-2, 4) \cup (4, \infty)$
Find all the real roots of the polynomial equation $x^3 - 2x^2 - 5x + 6 = 0$.
${1, 6, -1}$
${-1, -3, 2}$
${-2, 1, 3}$
Solve the polynomial inequality $x^2 - 5x + 4 < 0$.
$1 < x < 4$
$-1 < x < 4$
$x < 1$ or $x > 4$
$x < -4$ or $x > -1$
What is the solution set for the inequality $(x+2)(x-1)(x-3) \ge 0$?
${x | -2 \le x \le 1 }$
${x | x \le -2 \text{ or } 1 \le x \le 3 }$
${x | -2 \le x \le 1 \text{ or } x \ge 3 }$
${x | x \le -2 \text{ or } x \ge 3 }$
Which of the following polynomial equations has a root at $x=0$ with multiplicity 2, and a root at $x=5$ with multiplicity 1?
$2x^2 - 10x = 0$
$x^2 - 5x = 0$
$x^3 - 5x^2 = 0$
$x^3 + 5x^2 = 0$
The solution to the inequality $P(x) > 0$ is the set of all $x$-values for which the graph of $y=P(x)$ is...
...to the left of the y-axis.
...on the x-axis.
...below the x-axis.
...above the x-axis.
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