Question

ben & jerry are both trying to factor the same trinomial. only one of them factored it correctly. which student is correct? what mistake did the other student make? ben’s answer: (4x - 3)(x + 1) jerry’s answer: (4x + 3)(x - 1) trinomial: 4x² + x - 3

Explanation:

Step1: Expand Ben's factored form

To check if Ben's factoring is correct, we expand \((4x - 3)(x + 1)\). Using the distributive property (FOIL method):
First: \(4x\times x = 4x^{2}\)
Outer: \(4x\times1 = 4x\)
Inner: \(-3\times x=-3x\)
Last: \(-3\times1 = -3\)
Combine like terms: \(4x^{2}+4x - 3x-3=4x^{2}+x - 3\)

Step2: Expand Jerry's factored form

Now expand Jerry's factored form \((4x + 3)(x - 1)\):
First: \(4x\times x = 4x^{2}\)
Outer: \(4x\times(-1)=-4x\)
Inner: \(3\times x = 3x\)
Last: \(3\times(-1)=-3\)
Combine like terms: \(4x^{2}-4x + 3x-3=4x^{2}-x - 3\)

Step3: Compare with the trinomial

The trinomial to factor is \(4x^{2}+x - 3\). Ben's expansion matches the trinomial, while Jerry's expansion gives \(4x^{2}-x - 3\), which is different. So Jerry made a mistake in his factoring.

Answer:

Jerry made a mistake. When expanding Jerry's factored form \((4x + 3)(x - 1)\), we get \(4x^{2}-x - 3\), which does not match the trinomial \(4x^{2}+x - 3\). Ben's expansion of \((4x - 3)(x + 1)\) gives \(4x^{2}+x - 3\), which is correct.