Multiply or divide as indicated.
$\frac{12 n^2-27}{2 n^2-5 n+3} \div \frac{8 n^2+10 n-3}{n^2-4 n+3}= \square$ (Type your answer in factored form.)
Answer (1)
Rewrite the division as multiplication by the reciprocal.
Factor each of the polynomials completely.
Cancel out any common factors in the numerator and the denominator.
The simplified expression is 4 n − 1 3 ( n − 3 ) .
Explanation
Problem Analysis We are asked to simplify the expression 2 n 2 − 5 n + 3 12 n 2 − 27 ÷ n 2 − 4 n + 3 8 n 2 + 10 n − 3 . To do this, we will rewrite the division as multiplication by the reciprocal, factor each polynomial, and cancel common factors.
Rewrite as Multiplication First, rewrite the division as multiplication by the reciprocal: 2 n 2 − 5 n + 3 12 n 2 − 27 ÷ n 2 − 4 n + 3 8 n 2 + 10 n − 3 = 2 n 2 − 5 n + 3 12 n 2 − 27 × 8 n 2 + 10 n − 3 n 2 − 4 n + 3
Factor Polynomials Next, factor each of the polynomials:
12 n 2 − 27 = 3 ( 4 n 2 − 9 ) = 3 ( 2 n − 3 ) ( 2 n + 3 )
2 n 2 − 5 n + 3 = ( 2 n − 3 ) ( n − 1 )
n 2 − 4 n + 3 = ( n − 1 ) ( n − 3 )
8 n 2 + 10 n − 3 = ( 4 n − 1 ) ( 2 n + 3 )
Substitute Factored Forms Substitute the factored forms into the expression: ( 2 n − 3 ) ( n − 1 ) 3 ( 2 n − 3 ) ( 2 n + 3 ) × ( 4 n − 1 ) ( 2 n + 3 ) ( n − 1 ) ( n − 3 )
Cancel Common Factors Now, cancel out any common factors in the numerator and the denominator: ( 2 n − 3 ) ( n − 1 ) 3 ( 2 n − 3 ) ( 2 n + 3 ) × ( 4 n − 1 ) ( 2 n + 3 ) ( n − 1 ) ( n − 3 ) = 4 n − 1 3 ( n − 3 )
Final Answer Therefore, the simplified expression in factored form is 4 n − 1 3 ( n − 3 ) .
Examples
Rational expressions are useful in many areas of mathematics and physics. For example, when analyzing the motion of objects, rational functions can describe the relationship between distance, time, and speed. Simplifying these expressions allows for easier calculations and a better understanding of the relationships between these variables. In electrical engineering, rational functions are used to analyze circuits and determine the transfer function of a system. Simplifying these expressions helps engineers design and optimize circuits for specific applications.
