Question

operations with rational expressions: foundations
google classroom microsoft teams
\\(\frac{-12z^3 + 4z^2}{4z^2 + 16z}\\)
which expression is equivalent to the given expression, for all \\( z < -10 \\)?
choose 1 answer:
a \\(\frac{-11z}{4}\\)
b \\( -3z + \frac{z}{4} \\)
c \\(\frac{-3z^2 + z}{z + 4}\\)
d \\(\frac{3z^2 - z}{z + 4}\\)
show calculator

Explanation:

Step1: Factor numerator and denominator

Factor out common terms from numerator and denominator.
Numerator: \(-12z^3 + 4z^2 = 4z^2(-3z + 1)\)
Denominator: \(4z^2 + 16z = 4z(z + 4)\)
So the expression becomes \(\frac{4z^2(-3z + 1)}{4z(z + 4)}\).

Step2: Simplify the fraction

Cancel out common factors \(4z\) (note \(z
eq 0\), and \(z < -10\) ensures \(z
eq 0\) and \(z + 4
eq 0\) since \(z < -10\) implies \(z + 4 < -6
eq 0\)).
After canceling, we get \(\frac{z(-3z + 1)}{z + 4} = \frac{-3z^2 + z}{z + 4}\).

Answer:

C. \(\frac{-3z^2 + z}{z + 4}\)