Which of the following represents the completely factored form of the algebraic expression… — Armed Services Vocational Aptitude Battery (ASVAB)

Which of the following represents the completely factored form of the algebraic expression $30v^3 - 35v^2 - 15v$ ?

$5v(3v + 1)(2v - 3)$ Cevap
B
$5v(3v - 1)(2v + 3)$
C
$v(15v + 5)(2v - 3)$
D
$5v(v - 9)(v + 2)$

Cevap

The completely factored form is

5v(3v + 1)(2v - 3)

The correct answer accurately extracts the Greatest Common Factor of 5v from the original expression, leaving the non-monic trinomial 6v^2 - 7v - 3. Factoring this remaining trinomial by grouping correctly yields the binomials (3v + 1) and (2v - 3). Expanding these prime factors back confirms they produce the exact original expression.

Adım Adım Çözüm

Identify and factor out the greatest common factor (GCF) from all three terms in the polynomial

30v^3 - 35v^2 - 15v

5v(6v^2 - 7v - 3)

Factoring out the GCF simplifies the remaining polynomial and is a necessary first step for complete factorization.

Analyze the remaining quadratic trinomial

6v^2 - 7v - 3

to factor it by grouping. Find two numbers that multiply to

a \times c

(

6 \times -3 = -18

) and add to

b

(

-7

The two numbers are

-9

and

2

These numbers allow us to split the middle term (

-7v

) into two separate terms to enable factoring by grouping.

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

6v^2 - 9v + 2v - 3 = 3v(2v - 3) + 1(2v - 3) = (3v + 1)(2v - 3)

Grouping the terms reveals the common binomial factor of

(2v - 3)

Combine the GCF extracted in step 1 with the binomial factors found in step 3.

5v(3v + 1)(2v - 3)

This represents the original expression as a product of its simplest irreducible factors.

Anahtar Kavram

Factoring a polynomial by first extracting the greatest common factor (GCF), followed by factoring a non-monic quadratic trinomial using grouping.

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