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

Lucas groups the polynomial (12x^3 + 8x) + (–6x^2 – 4) to factor → 2 (2 x - 1) (3 x^2 + 2)

Explanation:

Factor the following:

12 x^3 - 6 x^2 + 8 x - 4

Hint: | Factor out the greatest common divisor of the coefficients of 12 x^3 - 6 x^2 + 8 x - 4.

Factor 2 out of 12 x^3 - 6 x^2 + 8 x - 4:

2 (6 x^3 - 3 x^2 + 4 x - 2)

Hint: | Factor pairs of terms in 6 x^3 - 3 x^2 + 4 x - 2 by grouping.

Factor terms by grouping. 6 x^3 - 3 x^2 + 4 x - 2 = (6 x^3 - 3 x^2) + (4 x - 2) = 3 x^2 (2 x - 1) + 2 (2 x - 1):

2 3 x^2 (2 x - 1) + 2 (2 x - 1)

Hint: | Factor common terms from 3 x^2 (2 x - 1) + 2 (2 x - 1).

Factor 2 x - 1 from 3 x^2 (2 x - 1) + 2 (2 x - 1):

Answer: 2 (2 x - 1) (3 x^2 + 2)

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.

Explanation:

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