For all x > 3, the expression \frac{x^3 - 2x^2 - 9x + 18}{x^2 - 5x + 6} - \frac{x^3 + 3x^2… — SAT

For all $x > 3$ , the expression $\frac{x^3 - 2x^2 - 9x + 18}{x^2 - 5x + 6} - \frac{x^3 + 3x^2 - 4x - 12}{x^2 + 5x + 6}$ is equivalent to the constant $k$ . What is the value of $k$ ?

Answer: 5

Answer

The value of the constant

k

is 5.

Factoring the numerators by grouping yields

x^3 - 2x^2 - 9x + 18 = (x-3)(x+3)(x-2)

and

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

. Dividing out their respective denominators

(x-2)(x-3)

and

(x+2)(x+3)

leaves the simplified linear expressions

x+3

and

x-2

. Subtracting these yields

(x+3) - (x-2) = 5

Step-by-Step Solution

Factor the numerator and denominator of the first rational expression and cancel common factors.

x + 3

To simplify the first fraction by reducing it to its lowest terms.

Factor the numerator and denominator of the second rational expression and cancel common factors.

x - 2

To simplify the second fraction by reducing it to its lowest terms.

Subtract the second simplified expression from the first simplified expression.

To find the constant value equivalent to the given difference of rational expressions, ensuring to distribute the negative sign to all parts of the subtracted binomial.

Key Concept

Simplifying rational expressions by factoring cubic polynomials by grouping and quadratic trinomials

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic