If the expression \frac{3x^3 - 8x^2 + kx - 6}{3x - 2} is equivalent to x^2 - 2x + 3 for al… — SAT

If the expression $\frac{3x^3 - 8x^2 + kx - 6}{3x - 2}$ is equivalent to $x^2 - 2x + 3$ for all $x \neq \frac{2}{3}$ , where $k$ is a constant, what is the value of $k$ ?

Answer: 13

Answer

By multiplying both sides of the equation by

3x - 2

, the rational expression simplifies to a polynomial identity:

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

. Expanding the right side yields

3x^3 - 8x^2 + 13x - 6

. Since the two polynomials are equivalent, their corresponding coefficients must be equal, meaning the coefficient of the linear term,

k

, must be equal to

13

Step-by-Step Solution

Multiply both sides of the equivalence by the denominator

(3x - 2)

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

To clear the fraction and align the polynomial expressions for coefficient comparison.

Expand the right side of the equation using the distributive property

3x^3 - 8x^2 + 13x - 6

To obtain the expanded form of the polynomial so that we can identify the coefficients of each term.

Equate the corresponding coefficients of the linear

x

terms on both sides of the equation

k = 13

Since the two expressions are equivalent for all values of

x

, their coefficients for each corresponding power of

x

must be equal.

Key Concept

Equivalent Algebraic Expressions

Question

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Step-by-Step Solution

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