Question

question

subtract: \\(\frac{3z - 1}{z + 4} - \frac{2}{z - 4}\\).

provide your answer below:

Explanation:

Find the least common denominator

Using the Least Common Denominator knowledge point

$$ \text{LCD} = (z + 4)(z - 4) $$

Convert to equivalent rational expressions

Using the Equivalent Rational Expressions knowledge point

$$ \frac{3z - 1}{z + 4} = \frac{(3z - 1)(z - 4)}{(z + 4)(z - 4)} $$

$$ \frac{2}{z - 4} = \frac{2(z + 4)}{(z + 4)(z - 4)} $$

Expand the numerators

Expand the polynomial products in each numerator:

$$ (3z - 1)(z - 4) = 3z^2 - 12z - z + 4 = 3z^2 - 13z + 4 $$

$$ 2(z + 4) = 2z + 8 $$

Subtract the rational expressions

Using the Subtracting Rational Expressions knowledge point

$$
LATEXBLOCK0
$$

Check for further simplification

Analyze the numerator \(3z^2 - 15z - 4\) for factorization:
The discriminant is:

$$ D = (-15)^2 - 4(3)(-4) = 225 + 48 = 273 $$

Since 273 is not a perfect square, the numerator cannot be factored over the integers. The expression is in simplest form.

Answer:

Subtract: \(\frac{3z - 1}{z + 4} - \frac{2}{z - 4}\).

Provide your answer below:
<blank>\(\frac{3z^2 - 15z - 4}{(z + 4)(z - 4)}\)</blank>