A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

Find a polynomial with integer coefficients that satisfy the given condition. R has degree 4 and zeros 1-2i and 1, with a zero of multiplicity 2.

  • This Question is Closed
  1. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    A zero of multiplicity 2 means there is one zero that appears twice.

  2. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Also, if a polynomial with integer coefficients has complex roots, then those roots must appear in pairs of complex conjugate roots.

  3. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    For example, if the polynomial has the root 3 + 5i, then it must also have 3 - 5i as a root.

  4. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i got everything it's just that i don't know how to multiply [x-(1+2i)] with [x-(1-2i)]

  5. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    This is one way of doing it: First, simplify the parentheses inside each expression. Then multiply them together using polynomial multiplication. That is, multiply every term of the first polynomial by every term of the second polynomial, then collect like terms.

  6. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    The other way of doing it is to turn the product into the product of a sum and a difference and end up with the difference of two squares. Then you simplify.

  7. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Are you ok with this problem now? Do you need more help?

  8. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \(\large y = [x-(1+2i)][x-(1-2i)](x - 1)^2\)

  9. campbell_st
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    looking at your complex zero you know \[x = 1 \pm 2i\] then rewriting you get \[x- 1 = \pm 2i\] square both sides of the equation \[(x -1)^2 = 4i^2\] or \[(x -1)^2 = -4\] then you can find the quadratic factor

  10. mathstudent55
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Method 1: \(\large y = \color{red}{[x-(1+2i)][x-(1-2i)]}(x - 1)^2\) \(\large y = \color{red}{(x-1- 2i)(x- 1+2i)}(x - 1)^2\) \(\large y = \color{red}{(x^2-x + 2xi -x +1 -2i -2xi + 2i -4i^2)}(x - 1)^2\) \(\large y = \color{red}{(x^2-2x +1+4)}(x - 1)^2\) \(\large y = \color{red}{(x^2-2x +5)}(x - 1)^2\) Method 2: \(\large y = \color{red}{[x-(1+2i)][x-(1-2i)]}(x - 1)^2\) \(\large y = \color{red}{[(x-1)-2i][(x-1)+2i]}(x - 1)^2\) \(\large y = \color{red}{[(x-1)^2-(2i)^2]}(x - 1)^2\) \(\large y = \color{red}{[x^2 - 2x + 1 - (-4)]}(x - 1)^2\) \(\large y = \color{red}{(x^2 - 2x + 5)}(x - 1)^2\) After doing the step above with either method 1 or method 2 or with @campbell_st 's method, you still need to square x - 1, and then multiply it all together.

  11. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.