Question

on a test, leo is asked to completely factor the polynomial ( 3x^3 - 3x + 5x^2 - 5 ). he uses double grouping to get ( (x^2 - 1)(3x + 5) ). why is leo’s work incorrect? choose two correct answers.
he has not factored completely, because the first binomial is not prime.
( x^2 - 1 ) can be factored further.
( (3x + 5) ) can be factored further.
leo should’ve started by factoring out a gcf of 5.
leo should’ve started by factoring out a gcf of 3.

Explanation:

Step1: Analyze initial factoring step

The polynomial is $3x^3 - 3x + 5x^2 - 5$. First, rearrange terms: $3x^3 + 5x^2 - 3x - 5$. The GCF of all terms is 1, but grouping can be used. However, note that $x^2-1$ is a difference of squares.

Step2: Factor $x^2-1$ further

$x^2 - 1 = (x-1)(x+1)$, so Leo's $(x^2-1)$ is not fully factored.

Step3: Verify GCF claim

The terms have no common GCF of 3 or 5, so those claims are incorrect. The binomial $(3x+5)$ is prime, so it cannot be factored further.

Answer:

1. He has not factored completely, because the first binomial is not prime.
2. $x^2 - 1$ can be factored further.