Question

multiply and simplify.
\frac{(2x - 5)(x + 7)}{x^3} \cdot \frac{x}{5 - 2x}
(no response) , \\ x \
eq 0, \frac{5}{2}
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1. - / 1 points

multiply and simplify.
\frac{2t^2 - t - 15}{t + 2} \cdot \frac{t^2 - t - 6}{t^2 - 6t + 9}
(no response) , \\ t \
eq 3, -2
resources
read it

Explanation:

First Problem (with \( x \)):

Step1: Factor and identify opposites

Notice that \( 5 - 2x = - (2x - 5) \). So rewrite the second fraction:
\( \frac{(2x - 5)(x + 7)}{x^3} \cdot \frac{x}{-(2x - 5)} \)

Step2: Cancel common factors

Cancel \( (2x - 5) \) and one \( x \) (since \( x^3 \) has \( x \) and the numerator has \( x \)):
\( \frac{(x + 7)}{x^2} \cdot \frac{1}{-1} \)

Step3: Multiply the remaining terms

Multiply the numerators and denominators:
\( \frac{-(x + 7)}{x^2} = \frac{-x - 7}{x^2} \) (or \( -\frac{x + 7}{x^2} \))

Step1: Factor all quadratics

Factor \( 2t^2 - t - 15 \): \( (2t + 5)(t - 3) \)
Factor \( t^2 - t - 6 \): \( (t - 3)(t + 2) \)
Factor \( t^2 - 6t + 9 \): \( (t - 3)^2 \)
So the expression becomes:
\( \frac{(2t + 5)(t - 3)}{t + 2} \cdot \frac{(t - 3)(t + 2)}{(t - 3)^2} \)

Step2: Cancel common factors

Cancel \( (t + 2) \), one \( (t - 3) \) from numerator and denominator, and another \( (t - 3) \):
\( \frac{(2t + 5)(t - 3)}{1} \cdot \frac{(t - 3)}{(t - 3)^2} \) simplifies to \( 2t + 5 \) (after canceling \( (t - 3) \) terms)

Answer:

\( -\frac{x + 7}{x^2} \) (or \( \frac{-x - 7}{x^2} \))

Second Problem (with \( t \)):