College

We appreciate your visit to Given that 5 is a zero of the function f x 4x 3 11x 2 54x 45 solve the polynomial equation 4x 3 11x 2. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Given that 5 is a zero of the function \( f(x) = 4x^3 - 11x^2 - 54x + 45 \), solve the polynomial equation:

\[ 4x^3 - 11x^2 - 54x + 45 = 0 \]

Answer :

Final answer:

To solve the polynomial equation 4x³ − 11x² − 54x + 45 = 0, we can determine if 5 is a zero by substituting it into the function. Then, using polynomial long division or synthetic division, we can find the quadratic equation resulting from dividing by (x - 5). Solving the resulting quadratic equation gives us the solutions for the original equation.


Explanation:

To solve the polynomial equation 4x³ – 11x² − 54x + 45 = 0, we are given that 5 is a zero of the function f(x) = 4x³ − 11x² − 54x + 45. Since 5 is a zero, it means when we substitute x = 5 into the function, the result will be 0. So, let's substitute:



  1. f(5) = 4(5)³ − 11(5)² − 54(5) + 45 = 500 - 275 - 270 + 45 = 0

Therefore, we have confirmed that 5 is indeed a zero of the given function. Now, we need to solve the equation 4x³ – 11x² − 54x + 45 = 0 completely. To do this, we can use polynomial long division or synthetic division to find the quadratic equation that results from dividing the original cubic equation by (x - 5). Dividing the equation, we get:



(x - 5)(4x² + 9x - 9) = 0



Now we solve the quadratic equation 4x² + 9x - 9 = 0. You can use factoring, the quadratic formula, or completing the square to solve it. By using the quadratic formula, we get:



  1. x = (-9 ± √(9² - 4(4)(-9))) / (2(4))

  2. x = (-9 ± √(81 + 144)) / 8

  3. x = (-9 ± √(225)) / 8

  4. x = (-9 ± 15) / 8

  5. x = (-9 + 15) / 8 or x = (-9 - 15) / 8

  6. x = 6 / 8 or x = -24 / 8

  7. x = 3/4 or x = -3

Therefore, the solutions to the equation 4x³ – 11x² − 54x + 45 = 0 are x = 5 (given) and x = 3/4 or x = -3.


Learn more about polynomial equations here:

https://brainly.com/question/30127172


Thanks for taking the time to read Given that 5 is a zero of the function f x 4x 3 11x 2 54x 45 solve the polynomial equation 4x 3 11x 2. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada

Other Questions