If the expression \frac{2x + 1}{x - 3} - \frac{x - 2}{x + 1} is rewritten in the equivalen… — SAT

If the expression $\frac{2x + 1}{x - 3} - \frac{x - 2}{x + 1}$ is rewritten in the equivalent form $a + \frac{bx + c}{x^2 - 2x - 3}$ for all $x > 3$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

A
-1
B
11
9Answer
D
19

Answer

The correct answer is 9. Finding a common denominator yields the combined numerator

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

. Expanding these terms gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

. Dividing

x^2 + 8x - 5

x^2 - 2x - 3

results in a quotient of

1

and a remainder of

10x - 2

. Thus, the expression is equivalent to

1 + \frac{10x - 2}{x^2 - 2x - 3}

, which gives

a = 1

b = 10

, and

c = -2

. The sum

a + b + c = 1 + 10 - 2 = 9

Step-by-Step Solution

Find a common denominator for the two rational expressions.

The common denominator is

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

. The combined expression is

\frac{(2x + 1)(x + 1) - (x - 2)(x - 3)}{x^2 - 2x - 3}

To combine the fractions, we need to express them with a common denominator.

Expand and simplify the numerator.

(2x + 1)(x + 1) = 2x^2 + 3x + 1

and

(x - 2)(x - 3) = x^2 - 5x + 6

. Subtracting them gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

Simplifying the numerator allows us to express the combined fraction as a single polynomial over the denominator.

Perform polynomial division or rewrite the numerator to match the form

a + \frac{bx + c}{x^2 - 2x - 3}

Rewriting the numerator:

x^2 + 8x - 5 = 1(x^2 - 2x - 3) + 10x - 2

. Thus, the expression becomes

1 + \frac{10x - 2}{x^2 - 2x - 3}

This separates the rational expression into a constant integer and a proper rational fraction.

Identify the values of

a

b

, and

c

, and find their sum.

a = 1

b = 10

, and

c = -2

. The sum is

a + b + c = 1 + 10 + (-2) = 9

We compare the coefficients from our result to the given form and calculate the requested sum.

Key Concept

Combining rational expressions and rewriting them using polynomial division or algebraic manipulation.

Estimated Time:2m 30s

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