Factor $f(x)$ into linear factors given that $k$ is a zero of $f(x)$.
$f(x)=2 x^3-5 x^2+1 x+2 ; k=1$

A. $f(x)=(x-1)(x-2)(2 x+1)$
B. $f(x)=(x-1)(x+1)(2 x-2)$
C. $f(x)=(x-1)(x+2)(2 x-1)$
D. $f(x)=(x+1)(x+2)(2 x-1)$

We factored the polynomial f ( x ) = 2 x 3 − 5 x 2 + x + 2 using polynomial long division since k = 1 is a zero, resulting in ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) . Thus, the correct option is A. f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) .
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Answered by Anonymous | 2025-07-04

Recognize that since k = 1 is a zero of f ( x ) , ( x − 1 ) must be a factor.
Perform polynomial long division to divide f ( x ) by ( x − 1 ) , resulting in 2 x 2 − 3 x − 2 .
Factor the quadratic 2 x 2 − 3 x − 2 into ( x − 2 ) ( 2 x + 1 ) .
Write the complete factorization: f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) .

f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) ​
Explanation

Problem Analysis We are given the polynomial f ( x ) = 2 x 3 − 5 x 2 + x + 2 and told that k = 1 is a zero of f ( x ) . This means that f ( 1 ) = 0 , and thus ( x − 1 ) is a factor of f ( x ) . Our goal is to factor f ( x ) into linear factors.

Polynomial Division Since ( x − 1 ) is a factor of f ( x ) , we can perform polynomial division to find the other factor. We can use synthetic division or polynomial long division. Let's use polynomial long division to divide 2 x 3 − 5 x 2 + x + 2 by ( x − 1 ) .

Result of Division Performing the polynomial long division, we get:


\t\t\t 2 x 2 − 3 x − 2 \t\t x − 1 | 2 x 3 − 5 x 2 + x + 2 \t\t\t 2 x 3 − 2 x 2 \t\t\t------------- \t\t\t\t − 3 x 2 + x \t\t\t\t − 3 x 2 + 3 x \t\t\t\t------------- \t\t\t\t\t − 2 x + 2 \t\t\t\t\t − 2 x + 2 \t\t\t\t\t------------- \t\t\t\t\t\t 0
So, 2 x 3 − 5 x 2 + x + 2 = ( x − 1 ) ( 2 x 2 − 3 x − 2 ) .

Factoring Quadratic Now we need to factor the quadratic 2 x 2 − 3 x − 2 . We are looking for two numbers that multiply to 2 ( − 2 ) = − 4 and add to − 3 . These numbers are − 4 and 1 . So we can rewrite the quadratic as:

2 x 2 − 4 x + x − 2 = 2 x ( x − 2 ) + 1 ( x − 2 ) = ( 2 x + 1 ) ( x − 2 ) .
Thus, 2 x 2 − 3 x − 2 = ( x − 2 ) ( 2 x + 1 ) .

Final Factorization Therefore, f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) . Comparing this with the given options, we see that the correct factorization is f ( x ) = ( x − 1 ) ( x − 2 ) ( 2 x + 1 ) .

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 systems, such as bridges or electrical circuits. By finding the roots of a polynomial, they can determine the critical points where the system might become unstable. Similarly, in economics, polynomial factorization can be used to model cost functions and optimize production processes. Understanding how to factor polynomials allows professionals to make informed decisions and design efficient systems.

Answered by GinnyAnswer | 2025-07-04