For all x \neq 2, the expression \frac{3x^2 + kx - 8}{x - 2} is equivalent to 3x + 10 + \f… — SAT

For all $x \neq 2$ , the expression $\frac{3x^2 + kx - 8}{x - 2}$ is equivalent to $3x + 10 + \frac{12}{x - 2}$ , where $k$ is a constant. What is the value of $k$ ?

Answer: 4

Answer

The value of the constant

k

4

The correct answer is

4

. By rewriting the right-hand side of the equation with a common denominator of

x - 2

, the expression becomes

\frac{(3x + 10)(x - 2) + 12}{x - 2}

. Expanding and simplifying the numerator yields

3x^2 + 4x - 8

. Comparing this to the numerator of the left-hand side,

3x^2 + kx - 8

, shows that the coefficient of the

x

term,

k

, must equal

4

Step-by-Step Solution

Set up the equation representing the equivalence of the two expressions.

\frac{3x^2 + kx - 8}{x - 2} = 3x + 10 + \frac{12}{x - 2}

This establishes the relationship that must hold for all

x \neq 2

Find a common denominator for the terms on the right-hand side.

3x + 10 + \frac{12}{x - 2} = \frac{(3x + 10)(x - 2)}{x - 2} + \frac{12}{x - 2}

To combine the terms into a single rational expression, they must have the same denominator.

Expand and simplify the numerator on the right-hand side.

(3x + 10)(x - 2) + 12 = 3x^2 + 4x - 8

Expanding the product of binomials and combining like terms yields the simplified numerator.

Equate the numerators of the simplified expressions to solve for

k

3x^2 + kx - 8 = 3x^2 + 4x - 8

, which means

k = 4

Since the denominators are equal, the numerators must be identical for all values of

x \neq 2

, meaning their corresponding coefficients must match.

Key Concept

Equivalent Algebraic Expressions

Alternative Method

An alternative method is to substitute a convenient value for

x

that is not equal to

2

. Substituting

x = 1

into both sides of the equivalence gives:

\frac{3(1)^2 + k(1) - 8}{1 - 2} = 3(1) + 10 + \frac{12}{1 - 2}

. Simplifying this yields

\frac{k - 5}{-1} = 13 - 12

, which simplifies to

-k + 5 = 1

. Solving for

k

results in

k = 4

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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