Chapter 11.1, Problem 53PPE

To determine

Determine whether the statement $\frac{a^2+6a+5}{2a+2}$ is always, sometimes, or never true for real numbers $a\text{and}b$ . Explain.

Answer to Problem 53PPE

  $\frac{a^2+6a+5}{2a+2}=\frac{a+5}{2}$ is always true.

Explanation of Solution

Given:

The expression: $\frac{a^2+6a+5}{2a+2}$

Concept Used:

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Calculation:

Find the factors of numerator and denominator of the expression $\frac{a^2+6a+5}{2a+2}$ and cancel the common factors from numerator and denominator.

  $\frac{a^2+6a+5}{2a+2}=\frac{(a+1)(a+5)}{2(a+1)}=\frac{a+5}{2}$

Simplified Expression $\frac{a^2+6a+5}{2a+2}=\frac{a+5}{2}$ is always true.

Thus, $\frac{a^2+6a+5}{2a+2}=\frac{a+5}{2}$ is always true.