If the expression \frac{6x^2 + 7x - 20}{2x - 3} is equivalent to ax + b + \frac{c}{2x-3} f… — SAT

If the expression $\frac{6x^2 + 7x - 20}{2x - 3}$ is equivalent to $ax + b + \frac{c}{2x-3}$ for all $x \neq 1.5$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

Answer: 15

Answer

The value of

a + b + c

is 15.

By dividing the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

, we find that the quotient is

3x + 8

and the remainder is

4

. Thus, the expression can be rewritten as

3x + 8 + \frac{4}{2x-3}

. Comparing this to the given expression

ax + b + \frac{c}{2x-3}

, we obtain

a = 3

b = 8

, and

c = 4

. Their sum is

3 + 8 + 4 = 15

Step-by-Step Solution

Set up the polynomial division of the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

Dividing

6x^2 + 7x - 20

2x - 3

To express the rational expression in terms of a polynomial quotient and a remainder.

Divide the first term of the numerator by the first term of the denominator to determine the first quotient term.

The first term is

3x

. Subtracting

3x(2x - 3)

from the numerator leaves

16x - 20

\frac{6x^2}{2x} = 3x

, and subtracting

6x^2 - 9x

from the polynomial leaves the next term to be divided.

Divide the leading term of the remaining expression by the leading term of the denominator to determine the constant term of the quotient.

The constant term is

8

. Subtracting

8(2x - 3)

from

16x - 20

leaves a remainder of

4

\frac{16x}{2x} = 8

, and subtracting

16x - 24

from

16x - 20

gives the final constant remainder.

Compare the quotient and remainder to the given form to identify

a

b

, and

c

a = 3

b = 8

, and

c = 4

The quotient is

3x + 8

and the remainder is

4

, which matches the form

ax + b + \frac{c}{2x-3}

Calculate the sum of

a

b

, and

c

3 + 8 + 4 = 15

The question asks for the value of

a + b + c

Key Concept

Polynomial division and rewrite of rational expressions

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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