Question

teresa is factoring this polynomial by grouping. which common factors should be used in the next step of factoring?
10x³+3x²−20x−6
(10x³+3x²)+(−20x−6)
○ x² and −2x
○ 2x² and −2x
○ x² and −2
○ 2x² and −2

Explanation:

Step1: Factor first group

For \(10x^{3}+3x^{2}\), the greatest common factor (GCF) of \(10x^{3}\) and \(3x^{2}\) is \(x^{2}\)? Wait, no, wait: \(10x^{3}=x^{2}\times10x\), \(3x^{2}=x^{2}\times3\)? Wait, no, actually, wait, maybe I made a mistake. Wait, \(10x^{3}+3x^{2}\), the GCF of 10 and 3 is 1? No, that can't be. Wait, no, the original polynomial is \(10x^{3}+3x^{2}-20x - 6\). When we group as \((10x^{3}+3x^{2})+(-20x - 6)\), let's factor the first group: \(10x^{3}+3x^{2}=x^{2}(10x + 3)\)? Wait, no, that doesn't seem right. Wait, maybe I messed up the grouping. Wait, maybe the correct grouping is \((10x^{3}-20x)+(3x^{2}-6)\). Let's try that. Then \(10x^{3}-20x = 10x(x^{2}-2)\), and \(3x^{2}-6=3(x^{2}-2)\). But the problem says the grouping is \((10x^{3}+3x^{2})+(-20x - 6)\). So let's factor the first group \(10x^{3}+3x^{2}\): the GCF of \(10x^{3}\) and \(3x^{2}\) is \(x^{2}\)? Wait, \(10x^{3}=x^{2}\times10x\), \(3x^{2}=x^{2}\times3\), so \(10x^{3}+3x^{2}=x^{2}(10x + 3)\). Now the second group: \(-20x - 6\). Let's factor out the GCF. The GCF of -20x and -6 is -2? Wait, \(-20x-6=-2(10x + 3)\). Ah! So for the first group \(10x^{3}+3x^{2}\), we can factor out \(x^{2}\)? Wait, no, wait: \(10x^{3}+3x^{2}=x^{2}(10x + 3)\), and the second group \(-20x - 6=-2(10x + 3)\). Wait, so the common factor for the first group is \(x^{2}\)? No, wait, that gives \(x^{2}(10x + 3)\) and the second group: \(-20x - 6=-2(10x + 3)\). So the common factors to factor out from each group are \(x^{2}\) (from the first group) and \(-2\) (from the second group)? Wait, no, let's check the options. The options are: \(x^{2}\) and \(-2x\); \(2x^{2}\) and \(-2x\); \(x^{2}\) and \(-2\); \(2x^{2}\) and \(-2\). Wait, maybe I made a mistake in the first group. Let's re-express the first group: \(10x^{3}+3x^{2}\). Wait, maybe the GCF is \(x^{2}\)? No, \(10x^{3}\) and \(3x^{2}\) have GCF \(x^{2}\)? Wait, 10 and 3 have no common factor other than 1, so the GCF of \(10x^{3}\) and \(3x^{2}\) is \(x^{2}\). Then the second group: \(-20x - 6\). Let's factor out -2: \(-20x - 6=-2(10x + 3)\). And the first group: \(10x^{3}+3x^{2}=x^{2}(10x + 3)\). So in the first group, we factor out \(x^{2}\), and in the second group, we factor out -2. So the common factors are \(x^{2}\) (from first group) and \(-2\) (from second group). Let's check the options: option C is \(x^{2}\) and \(-2\). Let's verify:

First group: \(10x^{3}+3x^{2}=x^{2}(10x + 3)\)

Second group: \(-20x - 6=-2(10x + 3)\)

Yes, that works. So the common factors are \(x^{2}\) and \(-2\).

Answer:

C. \(x^{2}\) and \(-2\)