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

Questions

1. 1.

   A polynomial $P(x)$ satisfies the equation $(x - 1)P(x) = x^4 + 2x^3 - x^2 + ax - 2$ for all real numbers $x$, where $a$ is a constant. What is the remainder when $P(x)$ is divided by $x + 3$?

   • A) $-4$
   • B) $-2$
   • C) $2$
   • D) $4$
2. 2.

   What is the quotient when the polynomial $2x^4 - 5x^3 - 3x^2 + 19x - 9$ is divided by $x^2 - 3x + 2$?

   • A) $2x^2 + x - 4$
   • B) $2x^2 - x - 4$
   • C) $2x^2 + x + 4$
   • D) $2x^2 - 11x + 28$
3. 3.

   Which statement correctly describes the end behavior of the polynomial function $f(x) = (3 - x)(2x + 1)^2(x - 4)$?

   • 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 11 when divided by $x - 3$ and a remainder of -4 when divided by $x + 2$. What is the remainder when $P(x)$ is divided by $x^2 - x - 6$?

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

   The polynomial $P(x) = x^3 + ax^2 + bx - 12$ has $x - 2$ as a factor, and dividing $P(x)$ by $x + 1$ yields a remainder of $-18$. What is the value of the constant $b$?

   • A) $-1$
   • B) 2
   • C) 4
   • D) 6