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

Which of the following expressions represents the completely factored form of the polynomial $3k^3 - 15k^2 - 42k$ ?

The completely factored form is

3k(k - 7)(k + 2)

The polynomial is completely factored by first extracting the Greatest Common Factor,

3k

, which simplifies the expression to

3k(k^2 - 5k - 14)

. Next, the remaining trinomial is factored by finding two numbers that multiply to

-14

and add to

-5

. These numbers are

-7

and

2

. Combining these gives the final fully factored form of

3k(k - 7)(k + 2)

Identify and factor out the Greatest Common Factor (GCF) from all three terms of the polynomial.

The GCF of

3k^3

-15k^2

, and

-42k

3k

. Factoring this out leaves

3k(k^2 - 5k - 14)

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

Factor the remaining quadratic trinomial,

k^2 - 5k - 14

Find two integers that multiply to

-14

(the constant term) and add to

-5

(the coefficient of the middle term). These integers are

-7

and

2

. This yields the binomial factors

(k - 7)

and

(k + 2)

These binomials represent the linear factors of the quadratic expression.

Combine the GCF and the binomial factors to write the fully factored expression.

3k(k - 7)(k + 2)

This represents the original polynomial as a product of its simplest prime factors.

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Estimated Time:1m 0s

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