Find the rational zeros of the polynomial function and write in factored form.

f(x) = x^3 + 2x^2 - 7x - 24

A. f(x) = (x-3)(x^2 + 8x + 5)
B. f(x) = (x-3)(x^2 + 5x + 8)
C. f(x) = (x+3)(x+5)(x+8)
D. f(x) = (x+3)(x-5)(x-8)

The rational zero of the polynomial function f ( x ) = x 3 + 2 x 2 − 7 x − 24 is x = 3 . The factored form of the polynomial is f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) . Therefore, the correct choice is Option B.
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Answered by Anonymous | 2025-07-04

Expand each of the given options and compare to the original polynomial.
Option 2, f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) , matches the original polynomial f ( x ) = x 3 + 2 x 2 − 7 x − 24 .
Find the zeros of the factored form. x − 3 = 0 gives x = 3 , and x 2 + 5 x + 8 has complex roots.
The rational zero is x = 3 , and the factored form is f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) .
The rational zero is x = 3 and the factored form is f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) ​

Explanation

Problem Analysis We are given the polynomial function f ( x ) = x 3 + 2 x 2 − 7 x − 24 and asked to find its rational zeros and factored form. We are also given four possible factored forms to choose from.

Solution Strategy To find the correct factored form, we can expand each option and compare it to the original polynomial.

Expanding the Options Let's expand each option:


Option 1: f ( x ) = ( x − 3 ) ( x 2 + 8 x + 5 ) = x 3 + 8 x 2 + 5 x − 3 x 2 − 24 x − 15 = x 3 + 5 x 2 − 19 x − 15 . This does not match the original polynomial.
Option 2: f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) = x 3 + 5 x 2 + 8 x − 3 x 2 − 15 x − 24 = x 3 + 2 x 2 − 7 x − 24 . This matches the original polynomial.
Option 3: f ( x ) = ( x + 3 ) ( x + 5 ) ( x + 8 ) = ( x + 3 ) ( x 2 + 13 x + 40 ) = x 3 + 13 x 2 + 40 x + 3 x 2 + 39 x + 120 = x 3 + 16 x 2 + 79 x + 120 . This does not match the original polynomial.
Option 4: f ( x ) = ( x + 3 ) ( x − 5 ) ( x − 8 ) = ( x + 3 ) ( x 2 − 13 x + 40 ) = x 3 − 13 x 2 + 40 x + 3 x 2 − 39 x + 120 = x 3 − 10 x 2 + x + 120 . This does not match the original polynomial.

Finding the Zeros Since Option 2 matches the original polynomial, we have f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) . To find the rational zeros, we set each factor to zero. The first factor gives us x − 3 = 0 , so x = 3 . For the quadratic factor x 2 + 5 x + 8 , we can use the quadratic formula to find its roots: x = 2 a − b ± b 2 − 4 a c ​ ​ = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( 8 ) ​ ​ = 2 − 5 ± 25 − 32 ​ ​ = 2 − 5 ± − 7 ​ ​ . Since the discriminant is negative, the roots are complex, not rational.

Final Answer Therefore, the only rational zero is x = 3 , and the factored form is f ( x ) = ( x − 3 ) ( x 2 + 5 x + 8 ) .


Examples
Polynomial functions are used in various fields such as physics, engineering, and economics to model real-world phenomena. For example, in physics, projectile motion can be modeled using a quadratic polynomial function. In economics, cost and revenue functions can be modeled using polynomial functions to analyze profit and break-even points. Understanding how to find the zeros and factored form of a polynomial is crucial for solving these types of problems.

Answered by GinnyAnswer | 2025-07-04