Polynomial functions (division, remainder and factor theorems, end behavior) Knowledge-Based Practice

Questions

1. 1.

   A cubic polynomial $P(x)$ with leading coefficient 1 satisfies $P(1) = P(2) = P(3) = 4$. What is the value of $P(4)$?

   • A) 4
   • B) 6
   • C) 10
   • D) 16
2. 2.

   What is the quotient when the polynomial $P(x) = 4x^4 - 6x^3 + 8x^2 - 5x + 3$ is divided by $2x - 1$?

   • A) $4x^3 - 4x^2 + 6x - 2$
   • B) $2x^3 - 2x^2 + 3x - 1$
   • C) $2x^3 - 4x^2 + 4x - 2$
   • D) $4x^3 - 2x^2 + 6x - 1$
3. 3.

   Which statement correctly describes the end behavior of the polynomial function $f(x) = -2x(x - 1)^2(3 - x)(x + 2)$?

   • A) As $x \to -\infty$, $f(x) \to \infty$, and as $x \to \infty$, $f(x) \to -\infty$
   • B) As $x \to -\infty$, $f(x) \to -\infty$, and as $x \to \infty$, $f(x) \to -\infty$
   • C) As $x \to -\infty$, $f(x) \to -\infty$, and as $x \to \infty$, $f(x) \to \infty$
   • D) As $x \to -\infty$, $f(x) \to \infty$, and as $x \to \infty$, $f(x) \to \infty$
4. 4.

   A polynomial $P(x)$ leaves a remainder of 7 when divided by $x - 2$ and a remainder of $-1$ when divided by $x + 2$. What is the remainder when $P(x)$ is divided by $x^2 - 4$?

   • A) $2x - 3$
   • B) $2x + 3$
   • C) $3x + 1$
   • D) $4x - 1$
5. 5.

   The polynomial $P(x) = 2x^3 + kx^2 - 7x + 6$ is divisible by $x - 2$. What is the remainder when $P(x)$ is divided by $x + 1$?

   • A) 3
   • B) 9
   • C) 13
   • D) $-1$

Answers and step-by-step solutions appear after you finish the test.