Multiply $\frac{x^2+6 x+8}{x^2+3 x-10} \cdot \frac{x^2+2 x-15}{x^2+3 x-4}$. State any restrictions on the variables.
A. $\frac{x^2-x-6}{x^2-3 x+2}$; the variable restrictions are $x \neq-5 x \neq-4, x \neq 1, x \neq 2$
B. $\frac{x^2+x-6}{x^2-3 x+2}$; the variable restrictions are $x \neq-5 x \neq-4, x \neq 1, x \neq 2$
C. $\frac{x^2-x+6}{x^2-3 x+2}$; the variable restrictions are $x \neq-5 x \neq-4, x \neq 1, x \neq 2$
D. $\frac{x^2-x-6}{x^2+3 x+2} ;$ the variable restrictions are $x \neq-5 x \neq-4, x \neq 1, x \neq 2$
Answer (2)
The simplified expression is x 2 − 3 x + 2 x 2 − x − 6 , and the variable restrictions are x = − 5 , x = − 4 , x = 1 , x = 2 . The correct answer choice is A.
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Factor the numerator and denominator of each rational expression.
Multiply the rational expressions and cancel common factors.
Determine the restrictions on the variable by finding the values that make the original denominators equal to zero.
The simplified expression is x 2 − 3 x + 2 x 2 − x − 6 and the restrictions are x = − 5 , x = − 4 , x = 1 , x = 2 .
Explanation
Understanding the problem We are asked to multiply two rational expressions and state the restrictions on the variable x. The expressions are x 2 + 3 x − 10 x 2 + 6 x + 8 and x 2 + 3 x − 4 x 2 + 2 x − 15 .
Factoring the expressions First, we need to factor each quadratic expression. Let's start with the first rational expression: x 2 + 3 x − 10 x 2 + 6 x + 8 The numerator factors as ( x + 2 ) ( x + 4 ) and the denominator factors as ( x − 2 ) ( x + 5 ) . So the first expression becomes: ( x − 2 ) ( x + 5 ) ( x + 2 ) ( x + 4 ) Now, let's factor the second rational expression: x 2 + 3 x − 4 x 2 + 2 x − 15 The numerator factors as ( x + 5 ) ( x − 3 ) and the denominator factors as ( x + 4 ) ( x − 1 ) . So the second expression becomes: ( x + 4 ) ( x − 1 ) ( x + 5 ) ( x − 3 )
Multiplying and simplifying Now we multiply the two rational expressions: ( x − 2 ) ( x + 5 ) ( x + 2 ) ( x + 4 ) ⋅ ( x + 4 ) ( x − 1 ) ( x + 5 ) ( x − 3 ) We can cancel the common factors ( x + 4 ) and ( x + 5 ) from the numerator and the denominator: ( x − 2 ) ( x + 2 ) ⋅ ( x − 1 ) ( x − 3 ) = ( x − 2 ) ( x − 1 ) ( x + 2 ) ( x − 3 ) Multiplying out the numerator and the denominator, we get: x 2 − x − 2 x + 2 x 2 − 3 x + 2 x − 6 = x 2 − 3 x + 2 x 2 − x − 6
Finding the restrictions The restrictions on x are the values that make the denominators of the original expression equal to zero. From the first expression, x 2 + 3 x − 10 = ( x + 5 ) ( x − 2 ) = 0 , so x = − 5 and x = 2 . From the second expression, x 2 + 3 x − 4 = ( x + 4 ) ( x − 1 ) = 0 , so x = − 4 and x = 1 . Therefore, the restrictions are x = − 5 , − 4 , 1 , 2 .
Final Answer Therefore, the simplified expression is x 2 − 3 x + 2 x 2 − x − 6 and the restrictions are x = − 5 , x = − 4 , x = 1 , x = 2 .
Examples
Rational expressions are useful in many real-world applications, such as calculating the cost of producing a certain number of items. For example, if the cost of producing x items is given by the expression x + 1 x 2 + 5 x + 6 , and the revenue generated from selling x items is given by x + 1 x 2 + 7 x + 12 , then the profit can be found by subtracting the cost from the revenue. Simplifying these rational expressions can help in determining the break-even point, where the profit is zero. Understanding how to manipulate and simplify these expressions is essential for making informed business decisions.
