Question
upstudy study bank question image url

Use the remainder theorem to determine if the given number \( c \) is a zero of the polynomial. \( f(x)=3 x^{3}-7 x^{2}-18 x+42 \) (a) \( c=-2 \) (b) \( c=-\sqrt{6} \)

Ask by Little Hilton. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

\( c = -2 \) is not a zero of the polynomial, and \( c = -\sqrt{6} \) is a zero.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To check if \( c \) is a zero of the polynomial \( f(x) = 3x^3 - 7x^2 - 18x + 42 \) using the remainder theorem, we need to evaluate \( f(c) \) for each value of \( c \). (a) For \( c = -2 \): \[ f(-2) = 3(-2)^3 - 7(-2)^2 - 18(-2) + 42 \] Calculating it step-by-step: \[ = 3(-8) - 7(4) + 36 + 42 \] \[ = -24 - 28 + 36 + 42 \] \[ = -52 + 78 \] \[ = 26 \]. Since \( f(-2) = 26 \neq 0 \), \( c = -2 \) is not a zero of the polynomial. (b) For \( c = -\sqrt{6} \): \[ f(-\sqrt{6}) = 3(-\sqrt{6})^3 - 7(-\sqrt{6})^2 - 18(-\sqrt{6}) + 42 \] Calculating it step-by-step: \[ = 3(-6\sqrt{6}) - 7(6) + 18\sqrt{6} + 42 \] \[ = -18\sqrt{6} - 42 + 18\sqrt{6} + 42 \] All terms involving \(\sqrt{6}\) cancel out, and the constant terms also cancel: \[ = 0 \]. Since \( f(-\sqrt{6}) = 0 \), \( c = -\sqrt{6} \) is a zero of the polynomial.

Related Questions

Latest Algebra Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy