Multiply.
$\frac{x^2+5 x+6}{x^2-4} \cdot \frac{x^2+3 x+2}{x^2+4 x+3}$
Answer (2)
Factor each quadratic expression.
Rewrite the expression with factored forms.
Cancel common factors.
The simplified expression is x − 2 x + 2 .
Explanation
Understanding the Problem We are given the expression x 2 − 4 x 2 + 5 x + 6 ⋅ x 2 + 4 x + 3 x 2 + 3 x + 2 to simplify. Our goal is to multiply the two rational expressions and simplify the result by factoring each quadratic expression and canceling common factors.
Factoring Quadratic Expressions First, we factor each of the quadratic expressions:
x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) ,
x 2 − 4 = ( x − 2 ) ( x + 2 ) ,
x 2 + 3 x + 2 = ( x + 1 ) ( x + 2 ) ,
x 2 + 4 x + 3 = ( x + 1 ) ( x + 3 ) .
Rewriting with Factored Forms Now, we rewrite the expression with the factored quadratics: ( x − 2 ) ( x + 2 ) ( x + 2 ) ( x + 3 ) ⋅ ( x + 1 ) ( x + 3 ) ( x + 1 ) ( x + 2 )
Canceling Common Factors Next, we cancel common factors in the numerator and denominator. We can cancel ( x + 2 ) , ( x + 3 ) , and ( x + 1 ) :
( x − 2 ) ( x + 2 ) ( x + 2 ) ( x + 3 ) ⋅ ( x + 1 ) ( x + 3 ) ( x + 1 ) ( x + 2 ) = x − 2 x + 2
Simplified Expression The simplified expression is x − 2 x + 2 .
Examples
Rational expressions are used in many areas of science and engineering. For example, in physics, they can be used to describe the relationship between voltage, current, and resistance in an electrical circuit. Simplifying rational expressions can help engineers to design more efficient circuits. Also, these expressions are used in calculating rates and proportions, such as determining the speed of a car or the concentration of a solution.
We factored the given quadratic expressions and rewrote the multiplication of the rational expressions in factored form. By canceling the common factors, we simplified the result to x − 2 x + 2 . This illustrates the importance of factoring in simplifying rational expressions effectively.
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