Question

factor the following by grouping.
$x^3 - 5x^2 + 7x - 35$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $x^3 - 5x^2 + 7x - 35 = \square$
b. the polynomial is prime.

Explanation:

Step1: Group the terms

We have the polynomial \(x^{3}-5x^{2}+7x - 35\). Group the first two terms and the last two terms: \((x^{3}-5x^{2})+(7x - 35)\)

Step2: Factor out the GCF from each group

From the first group \(x^{3}-5x^{2}\), the GCF is \(x^{2}\), so we get \(x^{2}(x - 5)\). From the second group \(7x-35\), the GCF is \(7\), so we get \(7(x - 5)\)

Step3: Factor out the common binomial factor

Now we have \(x^{2}(x - 5)+7(x - 5)\). The common binomial factor is \((x - 5)\), so we factor that out: \((x - 5)(x^{2}+7)\)
So the factored form of \(x^{3}-5x^{2}+7x - 35\) is \((x - 5)(x^{2}+7)\), so option A is correct (we need to fill in the box with the factored form).

Answer:

\((x - 5)(x^{2}+7)\)