To optimize a packaging design, a technician models the surface area using the algebraic e… — Armed Services Vocational Aptitude Battery (ASVAB)

To optimize a packaging design, a technician models the surface area using the algebraic expression $18xy^2 - 24xy - 10x$ . What is the fully factored equivalent of this expression?

$2x(3y - 5)(3y + 1)$ Cevap
B
$2x(3y + 5)(3y - 1)$
C
$x(6y - 10)(3y + 1)$
D
$2xy(9y - 17)$

Cevap

The fully factored equivalent is

2x(3y - 5)(3y + 1)

The expression is fully factored by first extracting the greatest common factor of

2x

, resulting in

2x(9y^2 - 12y - 5)

. Then, using the ac-method, the remaining non-monic quadratic trinomial is factored into

(3y - 5)(3y + 1)

, yielding the final correct product.

Adım Adım Çözüm

Identify and extract the greatest common factor (GCF) from all three terms.

2x(9y^2 - 12y - 5)

Extracting the GCF simplifies the polynomial and prepares the remaining quadratic trinomial for factoring.

Identify the components of the remaining quadratic trinomial

ay^2 + by + c

Here,

a = 9

b = -12

, and

c = -5

This establishes the values needed to use the ac-method for factoring non-monic quadratics.

Find two numbers that multiply to

ac

and add to

b

ac = 9 \times -5 = -45

. The two numbers that multiply to

-45

and add to

-12

are

-15

and

3

These numbers allow the middle term to be split, enabling factoring by grouping.

Rewrite the middle term and factor the trinomial by grouping.

9y^2 - 15y + 3y - 5 = 3y(3y - 5) + 1(3y - 5) = (3y - 5)(3y + 1)

This completes the factorization of the trinomial portion of the expression.

Combine the originally extracted GCF with the factored binomials.

2x(3y - 5)(3y + 1)

This yields the complete and fully factored form of the original polynomial.

Anahtar Kavram

Factoring Polynomials with a GCF

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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