Question

factor the polynomial completely. $4x^3 - 12x^2 + 7x - 21$ $4x^2(x - 3) + 7(x - 3)$ which is the completely factored polynomial? $(4x^2 + 7)(x - 3)$ $3x^2(x - 3)$ $4x^2(x + 4)$ $(4x^2 - 7)(x + 3)$

Explanation:

Step1: Group terms by pairs

Group first two and last two terms:
$4x^3 - 12x^2 + 7x - 21 = (4x^3 - 12x^2) + (7x - 21)$

Step2: Factor GCF from each group

Factor $4x^2$ from first group, 7 from second:
$(4x^3 - 12x^2) + (7x - 21) = 4x^2(x - 3) + 7(x - 3)$

Step3: Factor out common binomial

Factor $(x-3)$ from both terms:
$4x^2(x - 3) + 7(x - 3) = (4x^2 + 7)(x - 3)$

Step4: Check for further factoring

$4x^2+7$ cannot be factored over real numbers.

Answer:

$\boldsymbol{(4x^2 + 7)(x - 3)}$