Homework 10

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. If I have a polynomial $P(x)$ and I divide it by $x - c$, is the remainder $P(c)$?
  2. If I have a polynomial $P(x)$ and $P(c) = 0$, is $x - c$ is a factor?
  3. If I have a polynomial $P(x)$ and $x - c$ is a factor, is $P(c) = 0$?
  4. Two polynomials $P$ and $D$ are given. Use long division to divide $P(x)$ by $D(x)$ and express $P$ in the form \[P(x) = D(x)\cdot Q(x) + R(x)\]
    1. $P(x) = x^4 + 2x^3 - 10x, \qquad D(x) = x - 3$
    2. $P(x) = 18x^5 - 9x^4 + 3x^2 - 3, \qquad D(x) = 3x^2 - 3x + 1$
  5. Divide the following polynomials using long division. Write your answer as $P(x) = D(x) \cdot Q(x) + R(x)$.
    1. $\dfrac{4x^3 + 2x^2 - 2x - 3}{2x + 1}$
    2. $\dfrac{x^6 + x^4 + x^2 + 1}{x^2 + 1}$
  6. If I divide $x^3 + 3x^2 - 7x + 6$ by $x - 2$, what is the remainder? You can either justify with the Remainder Theorem or long division.
  7. If \[x^{201} - 2x^{199} + x^{52} - 2x^{32} + 3x + 1\] is divided by $x - 1$, what is the remainder?
  8. Show the number $c = -2$ is a zero of the polynomial $P(x) = x^3 + 2x^2 - 9x - 18$ by using the Factor Theorem.
  9. What is the definition of a complete factorization? In particular, what does a complete factorization depend on?
  10. Given the polynomial \[P(x) = x^{3}-x^{2}+2x-2\]
    1. Find a complete factorization over $\mathbb{R}$.
    2. Find a complete factorization over $\mathbb{C}$.
  11. Given $P(x) = x^3 - 3x^2 + 3x - 1$:
    1. Show $c = 1$ is a zero of $P(x)$.
    2. Find a complete factorization of $P(x)$ over $\mathbb{R}$.
  12. Given $P(x) = x^3 - x^2 - 11x + 15$:
    1. Show $c = 3$ is a zero of $P(x)$.
    2. Find a complete factorization of $P(x)$ over $\mathbb{R}$.
  13. Find a complete factorization over $\mathbb{C}$ of the following polynomials. Some zeros will be given. If not given, use factoring techniques from 1.3.
    1. $P(x) = x^2 + 25$
    2. $P(x) = x^2 + 2x + 2$
    3. $P(x) = x^3 - 7x^2 + 17x - 15,\qquad x = 3$ is a zero
    4. $P(x) = x^5 + x^3 + 8x^2 + 8$
    5. $P(x) = x^3 - x - 6, \qquad x = 2$ is a zero
    6. $P(x) = x^5 - 16x$
    7. $P(x) = 2x^3 - 8x^2 + 9x - 9, \qquad x = 3$ is a zero
    8. $P(x) = x^3 + 7x^2 + 18x + 18, \qquad x = -3$ is a zero
    9. $P(x) = x^5 + 6x^3 + 9x$
    10. $P(x) = x^3 - x^2 + x$
  14. Suppose a polynomial $P(x)$ has real coefficients. For example \[x^3 + 5x^2 - 8x + 1\] has all real coefficients but \[x^4 - ix^2 + 3 + i\] does not. If a polynomial has real coefficients, then if $P(x)$ factors as $(x - i)Q(x)$, what other factor must be present in $P(x)$?