Question

determining the next step
talia grouped the terms and factored out the gcf of the groups of the polynomial (15x^2 - 3x - 20x + 4). her work is shown below.

1. ((15x^2 - 3x) + (-20x + 4))
2. (3x(5x - 1) + 4(-5x + 1))

talia noticed that she does not have a common factor. what should she do?

• talia needs to leave the polynomial as is because it is prime and cannot be factored.
• talia needs to factor out a (3x) from the first group and a (4x) from the second group.
• talia needs to factor out a negative from one of the groups so the binomials will be the same.
• talia needs to apply the distributive property to get the expression ((3x + 4)(5x - 1)).

Explanation:

Step1: Analyze current factored form

Current form: $3x(5x - 1) + 4(-5x + 1)$

Step2: Identify matching binomial goal

We need identical binomials. Notice $-5x + 1 = -(5x - 1)$

Step3: Factor negative from second group

Rewrite second term: $4(-5x + 1) = -4(5x - 1)$
New expression: $3x(5x - 1) - 4(5x - 1)$

Step4: Evaluate options

Now we have a common binomial factor, so the issue was the sign of the binomial. Factoring out a negative from one group fixes this.

Answer:

Talia needs to factor out a negative from one of the groups so the binomials will be the same.