If the expression \frac{3x^2 + 10x - 8}{x + k} is equivalent to 3x - 2 for all x \neq -k, … — SAT

If the expression $\frac{3x^2 + 10x - 8}{x + k}$ is equivalent to $3x - 2$ for all $x \neq -k$ , where $k$ is a positive constant, what is the value of $k$ ?

Answer: 4

Answer

The value of the constant

k

is 4.

For the expressions to be equivalent for all values of

x

, the numerator

3x^2 + 10x - 8

must be equal to the product of the denominator

x + k

and the simplified quotient

3x - 2

. Expanding the product yields

3x^2 + (3k - 2)x - 2k

. Equating the constant terms on both sides gives

-8 = -2k

, which results in

k = 4

. Alternatively, equating the coefficients of the linear terms gives

10 = 3k - 2

, which also yields

k = 4

Step-by-Step Solution

Multiply both sides of the equivalence by the denominator

x + k

3x^2 + 10x - 8 = (3x - 2)(x + k)

To eliminate the fraction and set up a polynomial identity.

Expand the right side of the equation.

3x^2 + 10x - 8 = 3x^2 + (3k - 2)x - 2k

To express the right side in standard quadratic form for coefficient comparison.

Equate the constant terms to solve for

k

-8 = -2k \implies k = 4

Since the expressions are equivalent for all values of

x

, their corresponding coefficients and constants must be equal.

Key Concept

Equating coefficients of equivalent polynomial expressions

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