Multiply and simplify.
$\frac{y^2+4 y-21}{y^2+3 y-18} \cdot \frac{y^2-3 y-18}{y^2+10 y+21}$
The simplified product is $\square$ .
Answer (1)
Factor each quadratic expression.
Rewrite the expression with the factored quadratics.
Cancel out common factors in the numerator and the denominator.
The simplified product is y + 6 y − 6 .
Explanation
Understanding the Problem We are given the expression y 2 + 3 y − 18 y 2 + 4 y − 21 ⋅ y 2 + 10 y + 21 y 2 − 3 y − 18 to multiply and simplify. Our goal is to factor each quadratic expression, cancel common factors in the numerator and denominator, and write the simplified expression.
Factoring the First Quadratic First, let's factor the quadratic y 2 + 4 y − 21 . We are looking for two numbers that multiply to -21 and add to 4. These numbers are 7 and -3. So, y 2 + 4 y − 21 = ( y + 7 ) ( y − 3 ) .
Factoring the Second Quadratic Next, let's factor the quadratic y 2 + 3 y − 18 . We are looking for two numbers that multiply to -18 and add to 3. These numbers are 6 and -3. So, y 2 + 3 y − 18 = ( y + 6 ) ( y − 3 ) .
Factoring the Third Quadratic Now, let's factor the quadratic y 2 − 3 y − 18 . We are looking for two numbers that multiply to -18 and add to -3. These numbers are -6 and 3. So, y 2 − 3 y − 18 = ( y − 6 ) ( y + 3 ) .
Factoring the Fourth Quadratic Finally, let's factor the quadratic y 2 + 10 y + 21 . We are looking for two numbers that multiply to 21 and add to 10. These numbers are 7 and 3. So, y 2 + 10 y + 21 = ( y + 7 ) ( y + 3 ) .
Rewriting the Expression Now we rewrite the expression with the factored quadratics: ( y + 6 ) ( y − 3 ) ( y + 7 ) ( y − 3 ) ⋅ ( y + 7 ) ( y + 3 ) ( y − 6 ) ( y + 3 )
Canceling Common Factors Next, we cancel out common factors in the numerator and the denominator. We can cancel ( y − 3 ) , ( y + 7 ) , and ( y + 3 ) . ( y + 6 ) ( y − 3 ) ( y + 7 ) ( y − 3 ) ⋅ ( y + 7 ) ( y + 3 ) ( y − 6 ) ( y + 3 ) = y + 6 y − 6
Simplified Expression The simplified expression is y + 6 y − 6 .
Examples
Factoring and simplifying rational expressions is a fundamental skill in algebra, with applications in various fields. For instance, in physics, you might use it to simplify complex equations describing the motion of objects or the behavior of waves. In engineering, it can help in analyzing circuits or designing structures. Even in economics, simplifying expressions can aid in modeling supply and demand curves. Understanding how to manipulate these expressions allows you to solve problems more efficiently and gain deeper insights into the relationships between different variables.
