Question
upstudy study bank question image url

Write a polynomial \( f(x) \) that satisfies the given conditions. Polynomial of lowest degree with zeros of \( \frac{4}{3} \) (multiplicity 2 ) and \( -\frac{1}{2} \) (multiplicity 1 ) and with \( f(0)=-16 \).

Ask by Osborne Goodwin. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The polynomial is \( f(x) = -18 \left(x - \frac{4}{3}\right)^2 \left(x + \frac{1}{2}\right) \).

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

Bonus Knowledge

To construct the polynomial \( f(x) \), we start with its zeros and their multiplicities. The zeros given are \( \frac{4}{3} \) with multiplicity 2, and \( -\frac{1}{2} \) with multiplicity 1. Therefore, we can express the polynomial in factored form as: \[ f(x) = k \left( x - \frac{4}{3} \right)^2 \left( x + \frac{1}{2} \right) \] where \( k \) is a constant that we need to determine. Now, let's simplify the factors. Rewrite \( -\frac{4}{3} \) and \( \frac{1}{2} \) to eliminate the fractions. The polynomial becomes: \[ f(x) = k \left( 3x - 4 \right)^2 \left( 2x + 1 \right) \] Next, we need to find the value of \( k \) such that \( f(0) = -16 \). Substituting \( x = 0 \): \[ f(0) = k \left( 3(0) - 4 \right)^2 \left( 2(0) + 1 \right) = k (4)^2 (1) = 16k \] Setting this equal to \(-16\): \[ 16k = -16 \] Dividing both sides by 16 gives: \[ k = -1 \] Now we substitute \( k \) back into our polynomial: \[ f(x) = -1 \left( 3x - 4 \right)^2 \left( 2x + 1 \right) \] Now let's expand this polynomial: 1. Expand \( (3x - 4)^2 \): \[ (3x - 4)(3x - 4) = 9x^2 - 24x + 16 \] 2. Multiply by \( (2x + 1) \): \[ (9x^2 - 24x + 16)(2x + 1) = 18x^3 - 24x^2 + 9x^2 - 48x + 32 + 16 \] This yields: \[ 18x^3 - 24x^2 - 48x + 32 \] Now multiplying the expanded result by -1 gives: \[ f(x) = -18x^3 + 24x^2 + 48x - 32 \] Thus, the polynomial of lowest degree that satisfies the given conditions is: \[ \boxed{-18x^3 + 24x^2 + 48x - 32} \]

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