For all x \neq -2.5, the expression \frac{6x^2 + 17x + 8}{2x + 5} is equivalent to 3x + 1 … — SAT

For all $x \neq -2.5$ , the expression $\frac{6x^2 + 17x + 8}{2x + 5}$ is equivalent to $3x + 1 + \frac{a}{2x + 5}$ , where $a$ is a constant. What is the value of $a$ ?

Answer: 3

Answer

The value of

a

3

Multiplying both sides of the equivalent relation by

2x + 5

yields

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

. Expanding the right side gives

6x^2 + 17x + 5 + a

. Since the expressions are equivalent, the constant terms must be equal:

8 = 5 + a

, which simplifies to

a = 3

Step-by-Step Solution

Multiply both sides of the equation by

2x + 5

to clear the denominators.

6x^2 + 17x + 8 = (3x + 1)(2x + 5) + a

This clears the rational expressions so we can work with polynomials directly.

Expand the expression

(3x + 1)(2x + 5)

using the distributive property.

6x^2 + 17x + 8 = 6x^2 + 15x + 2x + 5 + a

To write the right side as a polynomial in standard form.

Combine like terms on the right side of the equation.

6x^2 + 17x + 8 = 6x^2 + 17x + (5 + a)

To group coefficients of like powers of

x

for easy comparison.

Equate the constant terms from both sides of the equation to solve for

a

8 = 5 + a

, which gives

a = 3

For the two polynomial expressions to be equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Rewriting rational expressions by clearing denominators or polynomial long division to find equivalent expressions.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic