Homework 9

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. A quadratic function $f(x)$ is given. Do the following four tasks for each function:
    • Find the maximum or minimum value of $f(x)$.
    • Describe the type of solutions of $f(x)$.
    1. $x^2 - 2x + 3$
    2. $-x^2 + 6x + 4$
    3. $-3x^2 + 6x - 2$
    4. $2x^2 + 12x + 10$
    5. $x^2 + 2x$
  2. Find a quadratic function that has a vertex of $(2, -3)$ and runs through the point $(3, 1)$.
    Hint: Use standard form to fill in the vertex. Then solve for $a$ with the point $(3, 1)$.
  3. Describe the end behavior for the following polynomials:
    1. $P(x) = 3x^3 - x^2$
    2. $P(x) = x^4 - 7x^2 + 5x + 5$
    3. $P(x) = -2x^5 - x^3$
    4. $P(x) = -3x^6 + x^4 + 18472x^2 + 24894837784$
  4. When graphing polynomials, are you allowed to have sharp corners or discontinuities?
  5. Sketch a rough graph of the following polynomials. Make sure the zeroes, end behavior, and zero multiplicity are displayed correctly.
    1. $P(x) = (x - 1)(x + 2)$
    2. $P(x) = -x(x-3)(x+2)$
    3. $P(x) = (x-1)^2(x+2)^3$
    4. $P(x) = x^3(x+2)(x-3)^2$
    5. $P(x) = x^4 - 2x^3 - 8x + 16$
    6. $P(x) = x^3 - x^2 - 6x$
    7. $P(x) = x^5 - 9x^3$
    8. $P(x) = 2x^3 - 2x^2 - 12x$
  6. Find a polynomial of degree 4 with zeroes $-1, 0, 2$ and $\sqrt{2}$.
  7. Given the function $P(x) = (x+1)(x+2)(x+3)$, explain why there is one local maximum and one local minimum.
  8. Suppose a polynomial $P(x)$ has zeroes at $x = -1, 0, 1$. If $P(x) > 0$ when $x \in (-1, 0)$ and $x \in (0, 1)$, what can we conclude about the multiplicity of $x = 0$?
  9. Write a polynomial with degree $5$ that has $5$ distinct real zeroes (okay if in factored form).
  10. Write a polynomial with degree 5 that has 2 distinct real zeroes.
  11. Write a polynomial with degree 86 with 3 distinct real zeroes.
  12. Write a polynomial with degree 4 with 2 distinct real zeroes.
  13. Write a polynomial with degree 6 with 5 distinct real zeroes.

    14. The rest of these problems will appear on next week's homework. Skip the rest of these for Homework 9.

  14. If I have a polynomial $P(x)$ and I divide it by $x - c$, is the remainder $P(c)$?
  15. If I have a polynomial $P(x)$ and $P(c) = 0$, is $x - c$ is a factor?
  16. If I have a polynomial $P(x)$ and $x - c$ is a factor, is $P(c) = 0$?
  17. 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. $P(x) = 18x^5 - 9x^4 + 3x^2 - 3, \qquad D(x) = 3x^2 - 3x + 1$
  18. 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. $\dfrac{x^6 + x^4 + x^2 + 1}{x^2 + 1}$
  19. 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.
  20. If \[x^{201} - 2x^{199} + x^{52} - 2x^{32} + 3x + 1\] is divided by $x - 1$, what is the remainder?
  21. Show the number $c = -2$ is a zero of the polynomial $P(x) = x^3 + 2x^2 - 9x - 18$ by using the Division algorithm.