Question

part d: choose the best method
decide whether gcf or grouping should be used first. then factor completely.

1. $6x^3 - 9x^2 + 12x$
2. $x^3 + 3x^2 + 4x + 12$

Explanation:

Step1: Identify GCF of terms

The terms are $6x^3$, $-9x^2$, $12x$. The GCF of coefficients 6, 9, 12 is 3; GCF of $x^3, x^2, x$ is $x$. Overall GCF: $3x$.
Factor out GCF:
$6x^3 - 9x^2 + 12x = 3x(2x^2 - 3x + 4)$

Step2: Check quadratic for factoring

Check if $2x^2 - 3x + 4$ can be factored. Discriminant: $\Delta = (-3)^2 - 4(2)(4) = 9 - 32 = -23 < 0$. No real factors.

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Step1: Group terms for factoring

Group terms with common factors:
$(x^3 + 3x^2) + (4x + 12)$

Step2: Factor GCF from each group

Factor $x^2$ from first group, 4 from second:
$x^2(x + 3) + 4(x + 3)$

Step3: Factor out common binomial

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

Step4: Check for further factoring

$x^2 + 4$ has discriminant $\Delta = 0^2 - 4(1)(4) = -16 < 0$, no real factors.

Answer:

1. $3x(2x^2 - 3x + 4)$
2. $(x + 3)(x^2 + 4)$