Homework 8

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 $P(-222) = 0$, what must be a factor of $P(x)$?
  3. If I have a polynomial $P(x)$ and $x + 3$ is a factor, is $P(3) = 0$?
  4. 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.
  5. If \[x^{201} - 2x^{199} + x^{52} - 2x^{32} + 3x + 1\] is divided by $x - 1$, what is the remainder?
  6. Suppose $P(x) = x^4 - 10x^2 + 9$. We know that $P(3) = 0$. What is the remainder when $P(x)$ is divided by $x - 3$?
  7. 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. $P(x) = 18x^5 - 9x^4 + 3x^2 - 3, \qquad D(x) = 3x^2 - 3x + 1$
  8. 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. $\dfrac{x^6 + x^4 + x^2 + 1}{x^2 + 1}$
      Hint Don't forget a placeholder for the divisor.
  9. A polynomial $P(x)$ is given, with a few zeros.
    Find a complete factorization over $\mathbb{R}$ using the division algorithm.
    You must use long division or no credit is received.
    1. $P(x) = x^3 + 2x^2 - 13x + 10, \qquad x = 1$
    2. $P(x) = 3x^4 - x^3 - 21x^2 -11x + 6, \qquad x = -1, -2$
    3. $P(x) = 4x^4 - 4x^3 - 3x^2 + 2x + 1, \qquad x = 1$
    4. $P(x) = x^4 - 5x^3 + 6x^2 + 4x - 8, \qquad x = 2$
  10. Find a polynomial of degree 4 with zeroes $-1, 0, 2$ and $\sqrt{2}$.
  11. Find a polynomial of degree 6 with zeroes $-3, 2$ and $4$.
  12. What is the definition of an irreducible polynomial?
  13. Write down the definition of zero from Section 3.2. I will ask you to recall this definition on next week's quiz!
  14. Suppose $P(x)$ factors into two irreducible quadratics and three linear factors over $\mathbb{R}$. What is the degree of $P(x)$?
  15. Given a complete factorization over $\mathbb{R}$, how do you find a complete factorization over $\mathbb{C}$?
  16. State whether each of the following statements are true or false.
    1. The point of Section 3.3 and Section 3.5 is to describe how polynomials factor.
    2. Every real number is a complex number.
    3. In the Real Factorization Theorem, it is possible for a polynomial to factor into only irreducible polynomials.
    4. The polynomial $P(x) = x^{234} - 4x^{22} - 4$ factors into $234$ linear factors over $\mathbb{C}$.
    5. Irreducible polynomials always factor into two linear factors over $\mathbb{R}$.
    6. The polynomial $P(x) = x^2 + 2$ factors into two linear factors over $\mathbb{R}$.
    7. The polynomial $P(x) = x^3 + x$ has one real zero and two complex zeros.
    8. Given a polynomial $P(x)$, the complete factorization over $\mathbb{R}$ is always the same as the complete factorization over $\mathbb{C}$.
    9. A complete factorization over $\mathbb{R}$ always results in linear factors.
    10. A polynomial of degree $n$ will always have $n$ real zeros.
    11. A polynomial of degree $n$ will always have $n$ complex zeros.
  17. Describe a polynomial of degree 5 with 3 linear factors and one irreducible factor.
  18. Suppose $P(x) = x^3 - x - 6$. Out of the numbers $-2, -1, 0, 1, 2$, which one(s) are zeros of $P(x)$?
  19. If $P(x)$ is degree two with two complex zeros $x = \dfrac{1 \pm i\sqrt{5}}{3}$, what does the complete factorization over $\mathbb{C}$ look like?
  20. A polynomial is given, sometimes with a few zeros. Do three things for each one:

    ① Determine the number of zeros, counting multiplicity.
    ② Find a complete factorization over $\mathbb{R}$.
    ③ Find a complete factorization over $\mathbb{C}$.

    1. $P(x) = 2x^2 - 2x + 3$
    2. $P(x) = x^3 + 2x^2 + 4x + 8$
    3. $P(x) = x^3 -7x^2 + 17x - 15 \qquad$ where $P(3) = 0$ (no need to verify).
    4. $P(x) = x^3 - x - 6 \qquad$ (use the zero from Problem 18).
    5. $P(x) = x^4 + 8x^2 + 16$
    6. $P(x) = 2x^6 - 4x^4 + 2x^2$
    7. $P(x) = 2x^3 - 8x^2 + 9x - 9 \qquad$ where $x = 3$ is a zero.

    21. Hint If you are struggling, see the example we did in class, which has a solution online now!