Question

cd: factor by gcf and grouping
factor the polynomial by grouping.
$4a^3 - 7a^2 + 4a - 7$
a $(4a + 7)(a^2 - 2)$
b $(7a - 9)(a^2 + 5)$
c $(7a + 9)(a^2 + 4)$
d $(4a - 7)(a^2 + 1)$

Explanation:

Step1: Group the terms

We have the polynomial \(4a^{3}-7a^{2}+4a - 7\). Group the first two terms and the last two terms: \((4a^{3}-7a^{2})+(4a - 7)\).

Step2: Factor out GCF from each group

For the first group \(4a^{3}-7a^{2}\), the GCF is \(a^{2}\), so we get \(a^{2}(4a - 7)\). For the second group \(4a - 7\), the GCF is \(1\), so we get \(1(4a - 7)\). Now the polynomial becomes \(a^{2}(4a - 7)+1(4a - 7)\).

Step3: Factor out the common binomial factor

We can see that \((4a - 7)\) is a common factor. Factoring it out, we get \((4a - 7)(a^{2}+1)\).

Answer:

D. \((4a - 7)(a^{2}+1)\)