Solve $\frac{x^2-81}{x+9} \div \frac{x^2+11 x+18}{x+2}$. Completely simplify your answer and state any restrictions on the variable.
A. $\frac{x-9}{x+9}, x \neq-9, x \neq-2$
B. $\frac{x-9}{x+2}, x \neq-9, x \neq-2$
C. $1, x \neq-9, x \neq-2$
D. $\frac{x+9}{x-9}, x \neq-9, x \neq-2$
Answer (2)
The simplified expression from the original division problem is x + 9 x − 9 with restrictions that x = − 9 and x = − 2 . This simplification came from factoring and canceling common terms. Therefore, the correct multiple choice answer is A: x + 9 x − 9 , x = − 9 , x = − 2 .
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Rewrite the division as multiplication by the reciprocal.
Factor the numerator and denominator.
Cancel common factors.
State the restrictions on the variable.
The simplified expression is x + 9 x − 9 , x = − 9 , x = − 2 .
Explanation
Understanding the problem We are asked to simplify the expression x + 9 x 2 − 81 ÷ x + 2 x 2 + 11 x + 18 and state any restrictions on the variable x .
Rewriting the division First, we rewrite the division as multiplication by the reciprocal: x + 9 x 2 − 81 ÷ x + 2 x 2 + 11 x + 18 = x + 9 x 2 − 81 ⋅ x 2 + 11 x + 18 x + 2
Factoring expressions Next, we factor the numerator x 2 − 81 as a difference of squares: x 2 − 81 = ( x − 9 ) ( x + 9 ) We also factor the denominator x 2 + 11 x + 18 . We are looking for two numbers that multiply to 18 and add to 11. These numbers are 2 and 9, so x 2 + 11 x + 18 = ( x + 2 ) ( x + 9 )
Substituting factored expressions Now we substitute these factorizations into the expression: x + 9 ( x − 9 ) ( x + 9 ) ⋅ ( x + 2 ) ( x + 9 ) x + 2
Canceling common factors We can cancel the common factors of ( x + 9 ) and ( x + 2 ) from the numerator and denominator, but we must remember that x = − 9 and x = − 2 because these values would make the denominators zero. After canceling, we have: x + 9 x − 9
Final Answer Therefore, the simplified expression is x + 9 x − 9 , with the restrictions x = − 9 and x = − 2 .
Examples
Rational expressions are used in many areas of science and engineering, such as in physics to describe the motion of objects or in electrical engineering to analyze circuits. For example, when analyzing the impedance of an electrical circuit, you might encounter a rational expression that needs to be simplified to understand the behavior of the circuit. Simplifying such expressions helps engineers design and optimize circuits for specific applications. Also, these expressions are useful in calculating rates and proportions, such as determining the efficiency of a machine or the concentration of a substance in a chemical reaction.
