One root of [tex]f(x)=x^3-4 x^2-20 x+48[/tex] is [tex]x=6[/tex]. What are all the factors of the function? Use the Remainder Theorem.
A. [tex](x+6)(x+8)[/tex]
B. [tex](x-6)(x-8)[/tex]
C. [tex](x-2)(x+4)(x-6)[/tex]
D. [tex](x+2)(x-4)(x+6)[/tex]
Answer (1)
Given the polynomial f ( x ) = x 3 − 4 x 2 − 20 x + 48 and the root x = 6 .
Divide f ( x ) by ( x − 6 ) to find the quadratic factor: x 2 + 2 x − 8 .
Factor the quadratic x 2 + 2 x − 8 into ( x + 4 ) ( x − 2 ) .
The complete factorization of f ( x ) is ( x − 6 ) ( x + 4 ) ( x − 2 ) . The factors are ( x − 2 ) ( x + 4 ) ( x − 6 ) .
Explanation
Problem Analysis We are given the polynomial f ( x ) = x 3 − 4 x 2 − 20 x + 48 and told that x = 6 is a root. This means that ( x − 6 ) is a factor of f ( x ) . We want to find all the factors of f ( x ) .
Polynomial Division Since we know that ( x − 6 ) is a factor, we can perform polynomial division to find the other factor, which will be a quadratic. Dividing f ( x ) by ( x − 6 ) , we get: ( x 3 − 4 x 2 − 20 x + 48 ) / ( x − 6 ) = x 2 + 2 x − 8
Factoring the Quadratic Now we need to factor the quadratic x 2 + 2 x − 8 . We are looking for two numbers that multiply to − 8 and add to 2 . These numbers are 4 and − 2 . So, we can factor the quadratic as: x 2 + 2 x − 8 = ( x + 4 ) ( x − 2 )
Complete Factorization Therefore, the complete factorization of f ( x ) is: f ( x ) = ( x − 6 ) ( x + 4 ) ( x − 2 )
Final Answer The factors of the function are ( x − 6 ) , ( x + 4 ) , and ( x − 2 ) .
Examples
Factoring polynomials is a fundamental concept in algebra and has many 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. Suppose you want to design a rectangular garden with a specific area, say x 3 − 4 x 2 − 20 x + 48 square feet. By factoring this polynomial, you can determine the possible dimensions (length and width) of the garden in terms of x . If one dimension is ( x − 6 ) feet, the other dimension would be ( x + 4 ) ( x − 2 ) feet, giving you options for the garden's layout.
