Which of the following is equivalent to the expression \frac{3x^2 - 5x - 4}{x - 2} for all… — SAT

Which of the following is equivalent to the expression $\frac{3x^2 - 5x - 4}{x - 2}$ for all $x \neq 2$ ?

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

The correct answer represents the equivalent expression obtained by performing polynomial division on the rational expression. Dividing the numerator

3x^2 - 5x - 4

by the denominator

x - 2

yields a quotient of

3x + 1

and a remainder of

-2

. This can be written in the form of the quotient plus the remainder over the divisor, resulting in

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

Divide the leading term of the numerator,

3x^2

, by the leading term of the denominator,

x

The first term of the quotient is

3x

This starts the polynomial long division process.

Multiply

3x

by the divisor

(x - 2)

and subtract the result from the numerator.

(3x^2 - 5x - 4) - (3x^2 - 6x) = x - 4

Subtracting the multiplied term helps find the remainder of the first division step.

Divide the leading term of the remaining expression,

x

, by the leading term of the divisor,

x

The second term of the quotient is

1

To continue the division process with the remaining terms.

Multiply

1

by the divisor

(x - 2)

and subtract the result from

x - 4

(x - 4) - (x - 2) = -2

This step determines the final remainder of

-2

because the degree of the remainder is now less than the degree of the divisor.

Express the final result as the sum of the quotient and the remainder divided by the divisor.

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

To construct the equivalent algebraic expression.

Equivalent Algebraic Expressions

Question