If the polynomial expression (3x - 5) is multiplied by the expression (Ax^2 + 2x - 7), the… — Armed Services Vocational Aptitude Battery (ASVAB)

If the polynomial expression $(3x - 5)$ is multiplied by the expression $(Ax^2 + 2x - 7)$ , the resulting simplified polynomial has an $x^2$ coefficient of $-19$ . What is the value of the constant $A$ ?

Answer: 5

Answer

The

x^2

terms in the product of

(3x - 5)

and

(Ax^2 + 2x - 7)

are generated by multiplying the

x

terms together

(3x \cdot 2x = 6x^2)

and by multiplying the constant by the

x^2

term

(-5 \cdot Ax^2 = -5Ax^2)

. Setting their sum equal to

-19x^2

gives the equation

6 - 5A = -19

. Solving for

A

yields

A = 5

Step-by-Step Solution

Identify combinations that yield an

x^2

term

The

x^2

terms come from multiplying

(3x)

(2x)

and

(-5)

(Ax^2)

To find the coefficient of a specific term without expanding the entire polynomial, only multiply the terms whose exponents sum to the target power.

Calculate the partial products

(3x)(2x) = 6x^2

and

(-5)(Ax^2) = -5Ax^2

These represent the only contributions to the

x^2

term in the final expanded polynomial.

Formulate an equation for the coefficient

6 - 5A = -19

The sum of the partial

x^2

coefficients must equal the total

x^2

coefficient given in the problem.

Solve the linear equation for

A

-5A = -25 \rightarrow A = 5

Isolating the variable

A

provides the final answer.

Key Concept

Multiplying Polynomials

Question

Answer

Step-by-Step Solution

Key Concept

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