For all x > 5, which of the following is equivalent to the expression \frac{3x^2 - 14x - 5… — SAT

For all $x > 5$ , which of the following is equivalent to the expression $\frac{3x^2 - 14x - 5}{3x + 1} - \frac{x^2 - 25}{x - 5}$ ?

-10

Factoring the numerators allows both rational expressions to be simplified. The first expression,

\frac{3x^2 - 14x - 5}{3x + 1}

, factors as

\frac{(3x + 1)(x - 5)}{3x + 1}

, which simplifies to

x - 5

since

x > 5

. The second expression,

\frac{x^2 - 25}{x - 5}

, factors as

\frac{(x - 5)(x + 5)}{x - 5}

, which simplifies to

x + 5

. Subtracting the second simplified expression from the first gives

(x - 5) - (x + 5) = x - 5 - x - 5 = -10

. Therefore, the equivalent expression is

-10

Factor the numerator of the first rational expression,

3x^2 - 14x - 5

(3x + 1)(x - 5)

To find common factors that can be simplified with the denominator

3x + 1

Simplify the first expression,

\frac{3x^2 - 14x - 5}{3x + 1}

, for

x > 5

x - 5

Since

x > 5

3x + 1 \neq 0

, allowing us to divide out the common factor

(3x + 1)

Factor the numerator of the second rational expression,

x^2 - 25

(x - 5)(x + 5)

Using the difference of squares identity,

a^2 - b^2 = (a - b)(a + b)

Simplify the second expression,

\frac{x^2 - 25}{x - 5}

, for

x > 5

x + 5

Since

x > 5

x - 5 \neq 0

, allowing us to divide out the common factor

(x - 5)

Subtract the second simplified expression from the first.

(x - 5) - (x + 5) = x - 5 - x - 5 = -10

Distributing the negative sign through the parentheses and combining like terms yields the final simplified equivalent value.

Simplifying rational expressions by factoring and performing polynomial operations.

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