Divide.

[tex]\frac{x^2-9}{x^2-3 x-4}-\frac{x^2+2 x-15}{x^2-x-12}[/tex]

A. [tex]\frac{x+3}{x+5}[/tex]

B. [tex]\frac{(x+3)(x-4)}{(x+1)(x+5)}[/tex]

C. [tex]\frac{(x+3)(x+5)}{x+1}[/tex]

D. [tex]\frac{(x+3)(x+3)}{(x+1)(x+5)}[/tex]

Factor the numerator and denominator of both rational expressions.
Rewrite the division as multiplication by the reciprocal.
Cancel out common factors.
Simplify the resulting expression: ( x + 1 ) ( x + 5 ) ( x + 3 ) ( x + 3 ) ​ ​

Explanation

Problem Analysis We are asked to divide two rational expressions. This involves factoring the numerators and denominators of both expressions, changing the division to multiplication by the reciprocal, and then canceling common factors.

Factoring Quadratics First, let's factor each quadratic expression:


Numerator of the first fraction: x 2 − 9 = ( x − 3 ) ( x + 3 ) Denominator of the first fraction: x 2 − 3 x − 4 = ( x − 4 ) ( x + 1 ) Numerator of the second fraction: x 2 + 2 x − 15 = ( x − 3 ) ( x + 5 ) Denominator of the second fraction: x 2 − x − 12 = ( x − 4 ) ( x + 3 )

Rewrite as Multiplication Now, rewrite the division as multiplication by the reciprocal:

x 2 − 3 x − 4 x 2 − 9 ​ ÷ x 2 − x − 12 x 2 + 2 x − 15 ​ = ( x − 4 ) ( x + 1 ) ( x − 3 ) ( x + 3 ) ​ × ( x − 3 ) ( x + 5 ) ( x − 4 ) ( x + 3 ) ​

Canceling Common Factors Next, cancel out the common factors:

( x − 4 ) ​ ( x + 1 ) ( x − 3 ) ​ ( x + 3 ) ​ × ( x − 3 ) ​ ( x + 5 ) ( x − 4 ) ​ ( x + 3 ) ​ = ( x + 1 ) ( x + 5 ) ( x + 3 ) ( x + 3 ) ​

Simplifying the Expression Finally, simplify the expression:

( x + 1 ) ( x + 5 ) ( x + 3 ) ( x + 3 ) ​ = ( x + 1 ) ( x + 5 ) ( x + 3 ) 2 ​ = x 2 + 6 x + 5 x 2 + 6 x + 9 ​

Final Result The simplified expression is ( x + 1 ) ( x + 5 ) ( x + 3 ) ( x + 3 ) ​ or x 2 + 6 x + 5 x 2 + 6 x + 9 ​ .

Examples
Rational expressions are used in various fields like physics and engineering to model complex relationships. For example, in electrical engineering, they can represent impedance in circuits. Simplifying these expressions helps engineers analyze and design circuits more efficiently. Similarly, in physics, they can describe the motion of objects under certain conditions, and simplification aids in understanding and predicting the behavior of these systems.

Answered by GinnyAnswer | 2025-07-08

To simplify the expression x 2 − 3 x − 4 x 2 − 9 ​ − x 2 − x − 12 x 2 + 2 x − 15 ​ , we factor both numerators and denominators, combine the fractions, and cancel common factors. The final result is ( x + 1 ) ( x + 5 ) ( x + 3 ) ( x + 3 ) ​ , which corresponds to answer choice D. Thus, the correct answer is D.
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Answered by Anonymous | 2025-07-29