Consider the polynomial 8x^2 - 10x - 3. When this expression is factored completely into t… — Armed Services Vocational Aptitude Battery (ASVAB)

Consider the polynomial $8x^2 - 10x - 3$ . When this expression is factored completely into the form $(4x + a)(2x - b)$ , where $a$ and $b$ are positive integers, what are the corresponding values for $a$ and $b$ ?

Answer:The value of

a

is 【1】 and the value of

b

is 【3】.

Answer

The value of a is 1 and the value of b is 3.

By applying the AC method, the quadratic

8x^2 - 10x - 3

can be rewritten as

8x^2 - 12x + 2x - 3

and factored by grouping into

4x(2x - 3) + 1(2x - 3) = (4x + 1)(2x - 3)

. Aligning this with

(4x + a)(2x - b)

directly shows that

a = 1

and

b = 3

Step-by-Step Solution

Identify the type of polynomial and the appropriate factoring method.

The polynomial

8x^2 - 10x - 3

is a non-monic quadratic trinomial (where the leading coefficient is not 1). We can use the ac-method.

The ac-method involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

Calculate the ac product and find the target sum.

The product is

a \times c = 8 \times (-3) = -24

. The target sum is the middle coefficient, which is

-10

These values are required to split the middle term for factoring by grouping.

Find two integers that multiply to -24 and add to -10.

The numbers

-12

and

2

meet these conditions, since

(-12) \times 2 = -24

and

(-12) + 2 = -10

This allows us to rewrite the middle term

-10x

without changing the value of the polynomial.

Rewrite the middle term and factor by grouping.

8x^2 - 12x + 2x - 3

. Grouping the terms yields

(8x^2 - 12x) + (2x - 3)

. Factoring out the greatest common factor from each group gives

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

Grouping transforms the four-term expression into a format where a common binomial factor can be extracted.

Factor out the common binomial factor and determine the final values.

Factoring out

(2x - 3)

leaves

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

. Comparing this to the given form

(4x + a)(2x - b)

reveals that

a = 1

and

b = 3

Matching the derived factorization with the required format identifies the specific constants requested by the problem.

Key Concept

Factoring non-monic quadratic trinomials using the AC method and grouping.

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