Concept Check Kurt factored 16a^2-40a-6a+15 by grouping and obtained (8a-3)(2a-5). Callie factored the same polynomial and gave an answer of (3-8a)(5-2a). Which answer is correct?

Key Concepts

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. This technique involves rearranging the terms into groups, factoring out the common factors from each group, and then factoring out the common binomial factor. Understanding this method is crucial for verifying the correctness of factored forms of polynomials.

Polynomial Equivalence

Polynomial equivalence refers to the idea that two different expressions can represent the same polynomial function. To determine if two factored forms are equivalent, one can expand both expressions and compare the resulting polynomials. This concept is essential for validating the correctness of the factored forms provided by Kurt and Callie.

Binomial Factoring

Binomial factoring involves expressing a polynomial as a product of two binomials. This is often used in quadratic expressions and requires an understanding of the relationships between the coefficients and roots of the polynomial. Recognizing how to manipulate and rearrange binomials is key to solving the problem and determining which factored form is accurate.