Questions: Reduce the rational expression to lowest terms. 5q^2 + 45q + 100 / 25q^2 - 25q - 500 5q^2 + 45q + 100 / 25q^2 - 25q - 500 =

Transcript text: 2.3 Graph sketching Reduce the rational expression to lowest terms. \[ \begin{array}{l} \frac{5 q^{2}+45 q+100}{25 q^{2}-25 q-500} \\ \frac{5 q^{2}+45 q+100}{25 q^{2}-25 q-500}= \end{array} \] $\square$

Solution

Solution Steps

To reduce the rational expression to its lowest terms, we need to factor both the numerator and the denominator and then cancel out any common factors.

Step 1: Factor the Numerator

The numerator $ 5q^2 + 45q + 100 $ can be factored as follows: \[ 5(q + 4)(q + 5) \]

Step 2: Factor the Denominator

The denominator $ 25q^2 - 25q - 500 $ can be factored as: \[ 25(q - 5)(q + 4) \]

Step 3: Simplify the Rational Expression

Now, we can write the rational expression as: \[ \frac{5(q + 4)(q + 5)}{25(q - 5)(q + 4)} \] We can cancel the common factor $ (q + 4) $: \[ \frac{5(q + 5)}{25(q - 5)} \] This simplifies to: \[ \frac{q + 5}{5(q - 5)} \]