Question

omar grouped the terms and factored the gcf out of the groups of the polynomial (3x^3 - 15x^2 - 4x + 20). his work is shown.
step 1: ((3x^3 - 15x^2) + (-4x + 20))
step 2: (3x^2(x - 5) + 4(-x + 5))
omar noticed that he does not have a common factor. which accurately describes what omar should do next?

• omar should realize that his work shows that the polynomial is prime.
• omar should go back and regroup the terms in step 1 as ((3x^3 - 15x^2) - (4x + 20)).
• in step 2, omar should factor only out of the first expression.
• omar should factor out a negative from one of the groups so the binomials will be the same.

Explanation:

To factor the polynomial \(3x^{3}-15x^{2}-4x + 20\) by grouping, after Step 1 \((3x^{3}-15x^{2})+(-4x + 20)\) and Step 2 \(3x^{2}(x - 5)+4(-x + 5)\), we notice that the binomials \((x - 5)\) and \((-x + 5)\) are related. If we factor out a negative from the second group \(-4x + 20\), we get \(-4(x - 5)\) (since \(-4x+20=-4(x - 5)\)). Then the expression becomes \(3x^{2}(x - 5)-4(x - 5)\), and we can factor out \((x - 5)\) to get \((x - 5)(3x^{2}-4)\).

• The first option is wrong because the polynomial is not prime (it can be factored further by adjusting the grouping).
• The second option's regrouping \((3x^{3}-15x^{2})-(4x + 20)\) is incorrect as \(4x + 20\) has a different sign relationship with the original terms.
• The third option is incorrect as factoring only out of the first expression does not help in factoring the entire polynomial.
• The fourth option is correct because factoring out a negative from one of the groups (the second group here) will make the binomials the same, allowing further factoring.

Answer:

Omar should factor out a negative from one of the groups so the binomials will be the same.