Perform the indicated operation of multiplication or division on the rational expressions and simplify.

$\frac{2 y^2-13 y-24}{y-4} \cdot \frac{y^2-6 y+8}{y-8}$

Factor the numerator of the first rational expression: 2 y 2 − 13 y − 24 = ( 2 y + 3 ) ( y − 8 ) .
Factor the numerator of the second rational expression: y 2 − 6 y + 8 = ( y − 4 ) ( y − 2 ) .
Rewrite the expression with factored numerators and cancel common factors: y − 4 ( 2 y + 3 ) ( y − 8 ) ​ ⋅ y − 8 ( y − 4 ) ( y − 2 ) ​ = ( 2 y + 3 ) ( y − 2 ) .
Expand the simplified expression: ( 2 y + 3 ) ( y − 2 ) = 2 y 2 − y − 6 . The final answer is 2 y 2 − y − 6 ​ .

Explanation

Understanding the Problem We are given the expression y − 4 2 y 2 − 13 y − 24 ​ ⋅ y − 8 y 2 − 6 y + 8 ​ . Our goal is to perform the multiplication and simplify the resulting rational expression.

Factoring the First Numerator First, we factor the quadratic expressions in the numerators of both rational expressions. For the first numerator, 2 y 2 − 13 y − 24 , we look for two numbers that multiply to 2 ⋅ − 24 = − 48 and add up to − 13 . These numbers are − 16 and 3 . So we can rewrite the quadratic as 2 y 2 − 16 y + 3 y − 24 . Factoring by grouping, we get 2 y ( y − 8 ) + 3 ( y − 8 ) = ( 2 y + 3 ) ( y − 8 ) .

Factoring the Second Numerator Next, we factor the second numerator, y 2 − 6 y + 8 . We look for two numbers that multiply to 8 and add up to − 6 . These numbers are − 4 and − 2 . So the quadratic factors as ( y − 4 ) ( y − 2 ) .

Rewriting the Expression Now we rewrite the original expression with the factored numerators: y − 4 ( 2 y + 3 ) ( y − 8 ) ​ ⋅ y − 8 ( y − 4 ) ( y − 2 ) ​

Canceling Common Factors We can now cancel out the common factors ( y − 8 ) and ( y − 4 ) from the numerator and denominator: y − 4 ​ ( 2 y + 3 ) ( y − 8 ) ​ ​ ⋅ y − 8 ​ ( y − 4 ) ​ ( y − 2 ) ​ = ( 2 y + 3 ) ( y − 2 )

Expanding the Expression Finally, we expand the expression ( 2 y + 3 ) ( y − 2 ) to get 2 y 2 − 4 y + 3 y − 6 = 2 y 2 − y − 6 .

Final Answer Therefore, the simplified expression is 2 y 2 − y − 6 .


Examples
Rational expressions are useful in various fields, such as physics and engineering, where they are used to model complex relationships between variables. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational expressions can appear in equations describing the motion of objects or the behavior of waves. By simplifying these expressions, physicists can gain a better understanding of the underlying phenomena and make more accurate predictions.

Answered by GinnyAnswer | 2025-07-07

To simplify the expression y − 4 2 y 2 − 13 y − 24 ​ ⋅ y − 8 y 2 − 6 y + 8 ​ , we first factor the numerators. After canceling common factors, we expand and find the final result as 2 y 2 − y − 6 .
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Answered by Anonymous | 2025-07-13