Which multiplication expression is equivalent to
[tex]$\begin{array}{c}
\frac{2 x^2-5 x-3}{4 x^2+12 x+5} \div \frac{3 x^2-11 x+6}{6 x^2+11 x-10} \\
\frac{(x-3)(2 x+1)}{(2 x+1)(2 x+5)} \cdot \frac{(x-3)(3 x-2)}{(2 x+5)(3 x-2)} \\
\frac{(2 x+1)(2 x+5)}{(x-3)(2 x+1)} \cdot \frac{(2 x+5)(3 x-2)}{(x-3)(3 x-2)} \\
\frac{(x-3)(2 x+1)}{(2 x+1)(2 x+5)} \cdot \frac{(2 x+5)(3 x-2)}{(x-3)(3 x-2)}
\end{array}$[/tex]
Answer (2)
Factor the quadratic expressions: 2 x 2 − 5 x − 3 = ( x − 3 ) ( 2 x + 1 ) , 4 x 2 + 12 x + 5 = ( 2 x + 1 ) ( 2 x + 5 ) , 3 x 2 − 11 x + 6 = ( x − 3 ) ( 3 x − 2 ) , 6 x 2 + 11 x − 10 = ( 2 x + 5 ) ( 3 x − 2 ) .
Rewrite the division as multiplication by the reciprocal: 4 x 2 + 12 x + 5 2 x 2 − 5 x − 3 ÷ 6 x 2 + 11 x − 10 3 x 2 − 11 x + 6 = ( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 ) .
Simplify the expression by canceling common factors.
The equivalent multiplication expression is: ( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 ) .
Explanation
Problem Analysis We are given a division of two rational functions and asked to find an equivalent multiplication expression. To do this, we will factor the quadratic expressions, rewrite the division as multiplication by the reciprocal, and simplify by canceling common factors. Finally, we will compare our result to the given options to find the equivalent expression.
Factoring Quadratics First, let's factor the quadratic expressions:
2 x 2 − 5 x − 3 = ( x − 3 ) ( 2 x + 1 )
4 x 2 + 12 x + 5 = ( 2 x + 1 ) ( 2 x + 5 )
3 x 2 − 11 x + 6 = ( x − 3 ) ( 3 x − 2 )
6 x 2 + 11 x − 10 = ( 2 x + 5 ) ( 3 x − 2 )
Rewriting Division as Multiplication Now, rewrite the original division as multiplication by the reciprocal:
4 x 2 + 12 x + 5 2 x 2 − 5 x − 3 ÷ 6 x 2 + 11 x − 10 3 x 2 − 11 x + 6 = ( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 )
Finding the Equivalent Expression Finally, we compare the simplified expression with the given multiplication expressions. The expression we derived is:
( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 )
This matches the last of the given options.
Final Answer Therefore, the equivalent multiplication expression is:
( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 )
Examples
Rational expressions are used in various fields, such as physics and engineering, to model relationships between different quantities. 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 effectively. Similarly, in physics, rational functions can appear in equations describing the motion of objects or the behavior of waves. Simplifying these expressions can help physicists understand and predict the behavior of physical systems.
The equivalent multiplication expression for the given division of rational functions is ( 2 x + 1 ) ( 2 x + 5 ) ( x − 3 ) ( 2 x + 1 ) ⋅ ( x − 3 ) ( 3 x − 2 ) ( 2 x + 5 ) ( 3 x − 2 ) . This was derived by factoring the quadratics and rewriting the division as multiplication. The resulting expression matches the second option provided in the question.
;
