If [tex]F(-2)[/tex] 0. What are all the factors of the function [tex]f(x)=x^3-2 x^2-68 x-120[/tex]? Use the Remainder Theorem.
[list=A]
[*] [tex](x+2)(x+60)[/tex]
[*] [tex](x-2)(x-60)[/tex]
[*] [tex](x-10)(x+2)(x+6)[/tex]
[*] [tex](x+10)(x-2)(x-6)[/tex]
[/list]
Answer (1)
Since f ( − 2 ) = 0 , ( x + 2 ) is a factor of f ( x ) .
Divide f ( x ) by ( x + 2 ) to get x 2 − 4 x − 60 .
Factor the quadratic x 2 − 4 x − 60 into ( x − 10 ) ( x + 6 ) .
The factors of f ( x ) are ( x + 2 ) ( x − 10 ) ( x + 6 ) , so the answer is ( x − 10 ) ( x + 2 ) ( x + 6 ) .
Explanation
Problem Analysis We are given the function f ( x ) = x 3 − 2 x 2 − 68 x − 120 and told that f ( − 2 ) = 0 . We want to find all the factors of f ( x ) using the Remainder Theorem.
Applying the Remainder Theorem The Remainder Theorem states that if f ( a ) = 0 , then ( x − a ) is a factor of f ( x ) . Since f ( − 2 ) = 0 , we know that ( x − ( − 2 )) = ( x + 2 ) is a factor of f ( x ) .
Polynomial Division To find the other factors, we can divide f ( x ) by ( x + 2 ) . Using polynomial division or synthetic division, we find that: ( x 3 − 2 x 2 − 68 x − 120 ) / ( x + 2 ) = x 2 − 4 x − 60
Factoring the Quadratic Now we need to factor the quadratic x 2 − 4 x − 60 . We are looking for two numbers that multiply to -60 and add to -4. These numbers are -10 and 6. So, we can factor the quadratic as: x 2 − 4 x − 60 = ( x − 10 ) ( x + 6 )
Final Factors Therefore, the factors of f ( x ) are ( x + 2 ) ( x − 10 ) ( x + 6 ) .
Selecting the Correct Option Comparing our factors with the given options, we see that the correct answer is ( x − 10 ) ( x + 2 ) ( x + 6 ) .
Examples
Factoring polynomials is a fundamental concept in algebra and has numerous real-world applications. For example, engineers use polynomial factorization to analyze the stability of structures, economists use it to model market behavior, and computer scientists use it to design efficient algorithms. Imagine you are designing a suspension bridge and need to ensure it can withstand certain loads. By expressing the load distribution as a polynomial, you can factor it to identify critical points where the stress is highest, allowing you to reinforce those areas and prevent structural failure. This ensures the safety and longevity of the bridge.
