Question

Lucas and Erick are factoring the polynomial

12x3 – 6x2 + 8x – 4. Lucas groups the polynomial (12x3 + 8x) + (–6x2 – 4) to factor. Erick groups the polynomial (12x3 – 6x2) + (8x – 4) to factor. Who correctly grouped the terms to factor? Explain.

Answer 1

The polynomial is:
12 x³ - 6 x² + 8 x - 4
Lucas:
( 12 x³ + 8 x ) + ( - 6 x² - 4 ) = 4 x ( 3 x² + 2 ) - 2 ( 3 x² + 2 ) =
= ( 3 x² + 2 ) ( 4 x - 2 ) = 2 ( 3 x² + 2 ) ( 2 x - 1 )
Erick:
( 12 x³ - 6 x² ) + ( 8 x - 4 ) = 6 x² ( 2 x - 1 ) + 4 ( 2 x - 1 ) =
= ( 2 x - 1 ) ( 6 x² + 4 ) = 2 ( 3 x² + 2 ) ( 2 x - 1 )
Answer:
Both students are correct. Each grouping leads to the same result.

Answer 2

Both students are correct because polynomials can be grouped in different ways to factor. Both ways result in a common binomial factor between the groups. Using the distributive property , this common binomial term can be factored out. Each grouping results in the same two binomial factors.