Question

jasmyne & drake are trying to factor out the gcf from the polynomial below. only one student has factored out the gcf correctly. which student is correct? what is wrong with the incorrect student’s work? \\(- 18x^{3} - 6x^{2} + 24x\\) \\(\text{jasmynes answer}\\) \\(- 3x(6x^{2} + 2x - 8)\\) \\(\text{drakes answer}\\) \\(- 6x(3x^{2} + x - 4)\\)

Explanation:

Step1: Find GCF of coefficients

Find GCF of -18, -6, 24. Factors: 18=2×3², 6=2×3, 24=2³×3. GCF of numbers: 6 (and consider sign, -6 as leading term is negative). For variables, GCF of \(x^3\), \(x^2\), \(x\) is \(x\). So GCF is \(-6x\) (or \(6x\) with sign adjusted, but factoring out \(-6x\) keeps signs consistent).

Step2: Check Jasmyne's work

She factored out \(-3x\). Let's expand her answer: \(-3x(6x^2 + 2x - 8) = -18x^3 -6x^2 +24x\) (wait, that's correct? Wait no, wait GCF of -18, -6, 24: GCF of 18,6,24 is 6. So GCF is 6x? Wait no, -18, -6, 24: the GCF of the absolute values is 6, and the variable part is x (lowest power). Now, the leading coefficient is negative, so we can factor out -6x or 6x. Let's check Drake's expansion: \(-6x(3x^2 + x -4) = -18x^3 -6x^2 +24x\) (correct). Jasmyne's: \(-3x(6x^2 + 2x -8) = -18x^3 -6x^2 +24x\) (also correct? Wait, no, because the GCF should be the greatest common factor. The GCF of -18, -6, 24 is 6 (since 18,6,24 are divisible by 6, and -18, -6, 24 are divisible by 6). So GCF is 6x? Wait, no, -18, -6, 24: GCF of coefficients (absolute values) is 6, variable is x. So GCF is 6x, but since the first term is negative, factoring out -6x is better, but factoring out -3x is also possible? Wait, no, GCF is the greatest common factor, so 6 is greater than 3. So Jasmyne factored out 3x (half of 6x), so she didn't factor out the GCF (she factored out a common factor, but not the greatest). Drake factored out 6x (with a negative sign), which is the GCF. Wait, let's check the GCF of -18, -6, 24:

Factors of 18: 1,2,3,6,9,18

Factors of 6:1,2,3,6

Factors of 24:1,2,3,4,6,8,12,24

Common factors: 1,2,3,6. Greatest is 6. So GCF of coefficients is 6. So GCF is 6x (since variable part is x). But since the first term is -18x³, factoring out -6x (which is -1×6x) gives:

\(-6x(3x² + x -4)\) (Drake's answer). Jasmyne factored out -3x, which is a common factor but not the greatest (since 6 is greater than 3). So Drake factored out the GCF, Jasmyne factored out a non - GCF (a common factor, but not the greatest). Wait, but when we expand Jasmyne's answer, it's correct, but the GCF should be the greatest. So the mistake is that Jasmyne did not factor out the greatest common factor (she factored out 3x instead of 6x), while Drake factored out the GCF (-6x).

Answer:

Drake is correct. Jasmyne did not factor out the greatest common factor (GCF); she factored out \(-3x\) instead of the GCF \(-6x\) (the GCF of \(-18\), \(-6\), and \(24\) for the coefficient part is \(6\), and for the variable part is \(x\), so the GCF is \(6x\) (or \(-6x\) to handle the negative leading coefficient), and she used \(3x\) which is not the greatest common factor).