Perform the operation and simplify.
$\begin{array}{c}
\frac{x^2+8 x+16}{x+2} \div \frac{x^2+6 x+8}{x^2-4} \\
\frac{(x-[?])(x+\square)}{x+}\end{array}$
Answer (1)
Rewrite the division as multiplication by the reciprocal.
Factor the polynomials: x 2 + 8 x + 16 = ( x + 4 ) 2 , x 2 − 4 = ( x − 2 ) ( x + 2 ) , x 2 + 6 x + 8 = ( x + 4 ) ( x + 2 ) .
Substitute the factored forms into the expression and cancel common factors.
The simplified expression is x + 2 ( x + 4 ) ( x − 2 ) , so the missing values are 2 and 4 .
Explanation
Understanding the Problem We are asked to simplify the expression x + 2 x 2 + 8 x + 16 ÷ x 2 − 4 x 2 + 6 x + 8 and express it in the form x + ( x − [ ?]) ( x + □ ) . This involves factoring polynomials and canceling common factors.
Rewriting the Expression First, we rewrite the division as multiplication by the reciprocal: x + 2 x 2 + 8 x + 16 × x 2 + 6 x + 8 x 2 − 4
Factoring the Polynomials Next, we factor the polynomials: x 2 + 8 x + 16 = ( x + 4 ) 2 \x 2 − 4 = ( x − 2 ) ( x + 2 ) \x 2 + 6 x + 8 = ( x + 4 ) ( x + 2 )
Substituting the Factored Forms Now, we substitute the factored forms into the expression: x + 2 ( x + 4 ) 2 × ( x + 4 ) ( x + 2 ) ( x − 2 ) ( x + 2 )
Canceling Common Factors We cancel common factors: x + 2 ( x + 4 ) ( x + 4 ) × ( x + 4 ) ( x + 2 ) ( x − 2 ) ( x + 2 ) = x + 2 ( x + 4 ) ( x − 2 )
Final Answer The simplified expression is x + 2 ( x + 4 ) ( x − 2 ) . Thus, the missing values are 4 and 2 .
Examples
Simplifying rational expressions is useful in various fields, such as physics and engineering, where complex equations can be simplified to make calculations easier. For example, when analyzing the motion of objects or designing electrical circuits, simplifying expressions can help in understanding the underlying relationships between variables and making predictions about the system's behavior. This skill is also fundamental in calculus when dealing with limits and derivatives of rational functions.
