For all x > 0 and x \neq 1, the expression \frac{x^{5/2} - x^{3/2} - x + 1}{x^{3/2} - x - … — SAT

For all $x > 0$ and $x \neq 1$ , the expression $\frac{x^{5/2} - x^{3/2} - x + 1}{x^{3/2} - x - x^{1/2} + 1}$ is equivalent to $x + x^a + b$ , where $a$ and $b$ are constants and $a < 1$ . What is the value of $2a + b$ ?

Answer: 2

Answer

The correct answer is 2.

Substituting

u = x^{1/2}

transforms the expression into

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

. Factoring the numerator by grouping yields

(u^3 - 1)(u^2 - 1)

, and factoring the denominator yields

(u^2 - 1)(u - 1)

. Canceling the common factor

(u^2 - 1)

leaves

\frac{u^3 - 1}{u - 1}

. Applying the difference of cubes formula to factor

u^3 - 1

(u - 1)(u^2 + u + 1)

allows us to cancel the

(u - 1)

term, leaving

u^2 + u + 1

. Re-substituting

u = x^{1/2}

results in the equivalent expression

x + x^{1/2} + 1

. Comparing this to

x + x^a + b

with

a < 1

gives

a = 0.5

and

b = 1

. Evaluating

2a + b

yields

2(0.5) + 1 = 2

Step-by-Step Solution

Introduce a variable substitution to eliminate fractional exponents.

Let

u = x^{1/2}

, which implies

x = u^2

. The expression becomes

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

Substitution simplifies working with fractional exponents and makes factoring patterns easier to identify.

Factor the numerator by grouping terms.

The numerator factors as

u^3(u^2 - 1) - (u^2 - 1) = (u^3 - 1)(u^2 - 1)

Grouping common terms helps simplify polynomials into products of lower-degree factors.

Factor the denominator by grouping terms.

The denominator factors as

u^2(u - 1) - (u - 1) = (u^2 - 1)(u - 1)

Factoring the denominator reveals common terms that can be canceled with the numerator.

Combine the factored forms and cancel the common factor.

The expression becomes

\frac{(u^3 - 1)(u^2 - 1)}{(u^2 - 1)(u - 1)} = \frac{u^3 - 1}{u - 1}

for

u \neq 1

Canceling

(u^2 - 1)

simplifies the rational expression since

x \neq 1

ensures

u^2 - 1 \neq 0

Simplify the remaining rational expression using the difference of cubes formula.

Factoring

u^3 - 1

(u - 1)(u^2 + u + 1)

and canceling

(u - 1)

yields

u^2 + u + 1

The difference of cubes formula allows cancellation of the linear term in the denominator.

Substitute back the original variable and determine the constants.

Substituting

u = x^{1/2}

back gives

x + x^{1/2} + 1

, meaning

a = 0.5

and

b = 1

This puts the expression in the final requested form

x + x^a + b

to find

a

and

b

Calculate the requested value

2a + b

2(0.5) + 1 = 2

This computes the final numeric answer requested by the question.

Key Concept

Simplifying rational expressions with fractional exponents by substitution and grouping

Question

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Step-by-Step Solution

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