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
$-2x^2 + x + 3$
$-8x^3 + 26x^2 - 19x + 3$ Cevap
D
$-8x^3 - 15x + 3$

Cevap

-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

Adım Adım Çözüm

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.

Anahtar Kavram

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

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Adım Adım Çözüm

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