Which shows one way to determine the factors of $x^3-12 x^2-2 x+24$ by grouping?

A. $x(x^2-12)+2(x^2-12)$
B. $x(x^2-12)-2(x^2-12)$
C. $x^2(x-12)+2(x-12)$
D. $x^2(x-12)-2(x-12)$

The correct way to determine the factors of the polynomial x 3 − 12 x 2 − 2 x + 24 by grouping is option D : x 2 ( x − 12 ) − 2 ( x − 12 ) , which expands to match the original polynomial.
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Answered by Anonymous | 2025-07-04

Expand each of the given options.
Compare the expanded form of each option with the original polynomial x 3 − 12 x 2 − 2 x + 24 .
The option whose expanded form matches the original polynomial is the correct grouping.
The correct grouping is x 2 ( x − 12 ) − 2 ( x − 12 ) ​ .

Explanation

Problem Analysis We are given the polynomial x 3 − 12 x 2 − 2 x + 24 and four possible groupings, and we want to determine which grouping is correct. The correct grouping should be algebraically equivalent to the original polynomial.

Expanding the Options Let's expand each of the given options and compare them to the original polynomial:


Option 1: x ( x 2 − 12 ) + 2 ( x − 12 ) = x 3 − 12 x + 2 x − 24 = x 3 − 10 x − 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 2: x ( x 2 − 12 ) − 2 ( x − 12 ) = x 3 − 12 x − 2 x + 24 = x 3 − 14 x + 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 3: x 2 ( x − 2 ) + 2 ( x − 12 ) = x 3 − 2 x 2 + 2 x − 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 4: x 2 ( x − 12 ) − 2 ( x − 12 ) = x 3 − 12 x 2 − 2 x + 24 . This is equal to x 3 − 12 x 2 − 2 x + 24 .

Expanding the Corrected Options However, the options provided in the question are:

x ( x 2 − 12 ) + 2 ( x 2 − 12 ) x ( x 2 − 12 ) − 2 ( x 2 − 12 ) x 2 ( x − 12 ) + 2 ( x − 12 ) x 2 ( x − 12 ) − 2 ( x − 12 )
Let's expand these options:
Option 1: x ( x 2 − 12 ) + 2 ( x 2 − 12 ) = x 3 − 12 x + 2 x 2 − 24 = x 3 + 2 x 2 − 12 x − 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 2: x ( x 2 − 12 ) − 2 ( x 2 − 12 ) = x 3 − 12 x − 2 x 2 + 24 = x 3 − 2 x 2 − 12 x + 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 3: x 2 ( x − 12 ) + 2 ( x − 12 ) = x 3 − 12 x 2 + 2 x − 24 . This is not equal to x 3 − 12 x 2 − 2 x + 24 .
Option 4: x 2 ( x − 12 ) − 2 ( x − 12 ) = x 3 − 12 x 2 − 2 x + 24 . This is equal to x 3 − 12 x 2 − 2 x + 24 .

Conclusion Therefore, the correct grouping is x 2 ( x − 12 ) − 2 ( x − 12 ) .

Examples
Factoring polynomials by grouping is a technique used to simplify complex expressions, which is crucial in many engineering and physics applications. For example, when analyzing circuits, engineers often encounter complex transfer functions that need to be simplified to understand the system's behavior. By factoring the numerator and denominator of the transfer function, they can identify poles and zeros, which determine the stability and frequency response of the circuit. Similarly, in physics, factoring polynomials can help solve equations related to quantum mechanics and electromagnetism, making it an essential skill for problem-solving in these fields.

Answered by GinnyAnswer | 2025-07-04