Find the completely factored form of the given polynomial:

-12x^2 + 30x + 72

Which of the following options shows the correct factorization?

The completely factored form is

-6(2x + 3)(x - 4)

The polynomial requires factoring out the Greatest Common Factor (GCF) of -6, which leaves the trinomial

2x^2 - 5x - 12

. Factoring this trinomial yields the binomials

(2x + 3)

and

(x - 4)

. Multiplying these by the GCF gives the fully factored expression.

Identify and factor out the Greatest Common Factor (GCF) from all terms in the polynomial

-12x^2 + 30x + 72

The GCF is -6. Factoring this out gives:

-6(2x^2 - 5x - 12)

Factoring out a negative GCF makes the leading coefficient of the trinomial positive, which simplifies the subsequent factoring steps.

Factor the remaining quadratic trinomial

2x^2 - 5x - 12

by finding two integers that multiply to

(2 \times -12) = -24

and add to the middle coefficient

-5

The two integers are -8 and 3.

These integers will be used to split the middle term to factor by grouping.

Rewrite the trinomial by splitting the middle term using the integers found, then factor by grouping.

2x^2 - 8x + 3x - 12 = 2x(x - 4) + 3(x - 4) = (2x + 3)(x - 4)

Grouping allows us to factor out common binomials to find the factored form of the trinomial.

Combine the GCF with the factored binomials to write the final completely factored expression.

-6(2x + 3)(x - 4)

A completely factored polynomial must include all extracted common factors alongside the irreducible binomial components.

Factoring polynomials using a Greatest Common Factor (GCF) and factoring non-monic trinomials.

Soru