For all x > 3, the expression \frac{2x^2 - 11x + 15}{x - 3} + \frac{3x^2 - 12}{x - 2} is e… — SAT

For all $x > 3$ , the expression $\frac{2x^2 - 11x + 15}{x - 3} + \frac{3x^2 - 12}{x - 2}$ is equivalent to $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Answer: 6

Answer

The value of

a + b

6

The correct answer is

6

. By factoring the numerators of both rational expressions, we can simplify them. The first term is

\frac{(2x-5)(x-3)}{x-3} = 2x-5

. The second term is

\frac{3(x-2)(x+2)}{x-2} = 3(x+2) = 3x+6

. Adding these simplified terms gives

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

. Comparing this to

ax+b

yields

a=5

and

b=1

, so

a+b = 5+1=6

Step-by-Step Solution

Factor the numerator of the first rational expression and simplify.

\frac{2x^2 - 11x + 15}{x - 3} = 2x - 5

Since

2x^2 - 11x + 15 = (2x - 5)(x - 3)

, the factor

(x - 3)

divides out for all

x > 3

Factor the numerator of the second rational expression and simplify.

\frac{3x^2 - 12}{x - 2} = 3x + 6

Since

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

, the factor

(x - 2)

divides out for all

x > 3

Combine the simplified terms by adding them.

5x + 1

Adding

(2x - 5)

and

(3x + 6)

yields

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

Identify the values of

a

and

b

and calculate their sum.

a = 5

b = 1

, and

a + b = 6

Comparing

5x + 1

with

ax + b

gives

a = 5

and

b = 1

, so

a + b = 5 + 1 = 6

Key Concept

Simplifying rational expressions by factoring the numerator and dividing out common factors.

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