Concept Check Kurt factored 16a²-40a-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?

검증된 단계별 안내

Start by writing down the original polynomial: \(16a^2 - 40a - 6a + 15\).

Combine like terms in the middle: \(-40a - 6a = -46a\), so the polynomial becomes \(16a^2 - 46a + 15\).

Check Kurt's factorization: \((8a - 3)(2a - 5)\). Multiply these binomials using the FOIL method: First, Outer, Inner, Last.

Check Callie's factorization: \((3 - 8a)(5 - 2a)\). Multiply these binomials using FOIL as well.

Compare the expanded forms from both factorizations to the original polynomial to determine if they are equivalent and thus correct.

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주요 개념

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Factoring by Grouping

Factoring by grouping involves splitting a polynomial into groups that have common factors, then factoring out the greatest common factor from each group. This method helps simplify complex polynomials into products of binomials or other polynomials, making it easier to solve or analyze them.

Equivalent Expressions and Factoring

Two factored forms are equivalent if they represent the same polynomial, even if their factors look different. Differences in sign or order of terms can produce different-looking factors that multiply to the same original expression, so recognizing equivalent expressions is key to verifying factorizations.

Distributive Property and Sign Changes

The distributive property allows multiplication over addition or subtraction, and changing the order or signs in factors can produce equivalent expressions. For example, factoring out a negative sign from a binomial changes its form but not its value, which explains why (8a-3)(2a-5) and (3-8a)(5-2a) can be equivalent.

Concept Check Kurt factored 16a2-40a-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?