A polynomial P(x) is defined as the product of the binomial (4x - 3) and the trinomial (-2… — Armed Services Vocational Aptitude Battery (ASVAB)

A polynomial $P(x)$ is defined as the product of the binomial $(4x - 3)$ and the trinomial $(-2x^2 + 5x - 1)$ . What is the fully expanded and simplified form of $P(x)$ ?

A
$-8x^3 + 14x^2 - 19x - 3$
B
$-2x^2 + x + 3$
$-8x^3 + 26x^2 - 19x + 3$ Answer
D
$-8x^3 - 15x + 3$

Answer

-8x^3 + 26x^2 - 19x + 3

The correct expression is found by successfully distributing both

4x

and

-3

across the trinomial

(-2x^2 + 5x - 1)

, maintaining correct exponents and signs, and finally combining the like terms:

20x^2

with

6x^2

, and

-4x

with

-15x

Step-by-Step Solution

Set up the multiplication using the distributive property.

4x(-2x^2 + 5x - 1) - 3(-2x^2 + 5x - 1)

Every term in the first polynomial must be multiplied by every term in the second polynomial.

Distribute

4x

through the trinomial.

4x(-2x^2) + 4x(5x) + 4x(-1) = -8x^3 + 20x^2 - 4x

Multiply coefficients and add exponents for the variable base.

Distribute

-3

through the trinomial.

-3(-2x^2) - 3(5x) - 3(-1) = 6x^2 - 15x + 3

Multiply the constant by each term, carefully applying integer sign rules.

Combine the results and group like terms.

-8x^3 + (20x^2 + 6x^2) + (-4x - 15x) + 3

Grouping terms with the same variable exponent prepares the expression for final simplification.

Add the coefficients of the like terms.

-8x^3 + 26x^2 - 19x + 3

This yields the polynomial in standard form.

Key Concept

Multiplying polynomials by distributing every term in the first factor to every term in the second, then combining like terms.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic