Perform the indicated operations and reduce to lowest terms. Assume that no denominator has a value of zero.
[tex]\frac{5 x^2-45}{x^6} \div \frac{x-3}{x^4}[/tex]
Answer (2)
Rewrite the division as multiplication by the reciprocal: x 6 5 x 2 − 45 ⋅ x − 3 x 4 .
Factor the numerator: 5 x 2 − 45 = 5 ( x − 3 ) ( x + 3 ) .
Cancel the common factors ( x − 3 ) and x 4 .
The simplified expression is x 2 5 ( x + 3 ) .
Explanation
Understanding the Problem We are asked to simplify the expression x 6 5 x 2 − 45 ÷ x 4 x − 3 . This involves factoring, dividing rational expressions, and reducing to lowest terms.
Rewrite as Multiplication First, we rewrite the division as multiplication by the reciprocal: x 6 5 x 2 − 45 ÷ x 4 x − 3 = x 6 5 x 2 − 45 ⋅ x − 3 x 4
Factor the Numerator Next, we factor the numerator of the first fraction. We can factor out a 5, and then we have a difference of squares: 5 x 2 − 45 = 5 ( x 2 − 9 ) = 5 ( x − 3 ) ( x + 3 )
Substitute Factored Form Now we substitute the factored form into the expression: x 6 5 ( x − 3 ) ( x + 3 ) ⋅ x − 3 x 4
Cancel Common Factors We can now cancel common factors. We have a factor of ( x − 3 ) in both the numerator and denominator, and we can also cancel x 4 from the numerator and denominator: x 6 5 ( x − 3 ) ( x + 3 ) ⋅ x − 3 x 4 = x 2 5 ( x + 3 )
Simplified Expression Thus, the simplified expression is x 2 5 ( x + 3 ) . We can also distribute the 5 to write it as x 2 5 x + 15 .
Examples
Understanding how to simplify rational expressions is crucial in many areas of mathematics and physics. For instance, when analyzing electrical circuits, you might encounter complex fractions involving impedances. Simplifying these expressions allows you to determine the overall impedance of the circuit and understand how current flows through it. Similarly, in physics, simplifying rational expressions can help in analyzing wave functions or solving equations related to motion and forces.
The expression x 6 5 x 2 − 45 ÷ x 4 x − 3 simplifies to x 2 5 ( x + 3 ) . This is done by rewriting the division as multiplication by the reciprocal, factoring, and canceling common factors. The final expression can also be represented as x 2 5 x + 15 .
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