Problem 9 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\frac{x^{2}+9 x+14}{x+7} \cdot \frac{1}{x+2}$$

Step-by-Step Solution

Answer

The result of the multiplication is 1.

Step 1: Factoring the Quadratic

First, look at the quadratic \(x^{2}+9x+14\) in the first fraction. It can be factored into \((x+7)\times(x+2)\) by using factoring by grouping.

Step 2: Simplify the Fractions

Now, replace the quadratic in the first fraction with its factored form. The fractions become \(\frac{(x+7)(x+2)}{x+7} \cdot \frac{1}{x+2}\). Afterwards, simplify the expressions by cancelling out common terms in numerators and denominators, we get \(1 \cdot 1 = 1\).

Step 3: Apply the Multiplication rule for Fractions

As the last step, multiply the simplified fractions using the rule \((a/b) \cdot (c/d) = a \cdot c / b \cdot d\). As both of fractions are 1, multiplication results in \(1/1 = 1\).