If the polynomial 12x^2 + 10x - 8 is factored completely into the form k(ax - 1)(bx + c), … — ACT

If the polynomial $12x^2 + 10x - 8$ is factored completely into the form $k(ax - 1)(bx + c)$ , where $k$ , $a$ , $b$ , and $c$ are positive integers, what is the value of $k + a + b + c$ ?

Answer: 11

Answer

The value of

k + a + b + c

11

To factor the polynomial

12x^2 + 10x - 8

completely, we first factor out the greatest common factor of

2

, yielding

2(6x^2 + 5x - 4)

. Next, we factor the quadratic trinomial

6x^2 + 5x - 4

by finding two numbers that multiply to

6 \times (-4) = -24

and add to

5

. These numbers are

8

and

-3

. Splitting the linear term and factoring by grouping gives

6x^2 + 8x - 3x - 4 = 2x(3x + 4) - 1(3x + 4) = (2x - 1)(3x + 4)

. The completely factored expression is

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

. Comparing this with

k(ax - 1)(bx + c)

where

k, a, b, c

are positive integers, we determine that

k = 2

a = 2

b = 3

, and

c = 4

. Summing these values gives

2 + 2 + 3 + 4 = 11

Step-by-Step Solution

Factor out the greatest common factor (GCF) from the terms of the polynomial.

2(6x^2 + 5x - 4)

Factoring out the greatest common factor simplifies the coefficients, making the quadratic trinomial easier to factor.

Find two integers that multiply to

ac = 6 \times (-4) = -24

and add to

b = 5

The two numbers are

8

and

-3

These integers are needed to split the linear term in order to factor the quadratic by grouping.

Rewrite the middle term and factor the trinomial by grouping.

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

Rewriting the trinomial as

6x^2 + 8x - 3x - 4

allows grouping of the first two terms

2x(3x + 4)

and the last two terms

-1(3x + 4)

to extract the common binomial factor.

Combine the factors and match the coefficients to the form

k(ax - 1)(bx + c)

k = 2

a = 2

b = 3

, and

c = 4

The completely factored expression is

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

. Matching this to the given template where all constants are positive integers yields

k = 2

a = 2

b = 3

, and

c = 4

Calculate the sum of the constants.

11

Adding the values gives

2 + 2 + 3 + 4 = 11

Key Concept

Factoring quadratic trinomials of the form

ax^2 + bx + c

after removing a greatest common factor.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic