\frac{x^2-8 x+15}{5 x^2+10 x}+\frac{x^2-9}{10 x^2} \times \frac{x^2+5 x+6}{2 x-10}
Answer (2)
The expression is simplified step by step by factoring polynomials, substituting them into the original expression, finding a common denominator, and expanding the terms. Finally, the expression is simplified to 20 x 2 ( x − 5 ) ( x + 2 ) ( x − 3 ) ( x 4 + 14 x 3 − 3 x 2 + 160 x + 36 ) . This process demonstrates how to properly manipulate rational expressions in algebra.
;
Factor the polynomials in the expression.
Substitute the factored forms into the original expression and multiply.
Find a common denominator and combine the terms.
Simplify the expression.
The simplified expression is 20 x 2 ( x − 5 ) ( x + 2 ) ( x − 3 ) ( x 4 + 14 x 3 − 3 x 2 + 160 x + 36 ) .
Explanation
Factoring the Polynomials We are asked to simplify the expression 5 x 2 + 10 x x 2 − 8 x + 15 + 10 x 2 x 2 − 9 × 2 x − 10 x 2 + 5 x + 6 First, we factor all the polynomials in the expression. x 2 − 8 x + 15 = ( x − 3 ) ( x − 5 ) 5 x 2 + 10 x = 5 x ( x + 2 ) x 2 − 9 = ( x − 3 ) ( x + 3 ) x 2 + 5 x + 6 = ( x + 2 ) ( x + 3 ) 2 x − 10 = 2 ( x − 5 )
Substituting and Multiplying Substituting the factored forms into the original expression, we get 5 x ( x + 2 ) ( x − 3 ) ( x − 5 ) + 10 x 2 ( x − 3 ) ( x + 3 ) × 2 ( x − 5 ) ( x + 2 ) ( x + 3 ) Now, we multiply the second term: 5 x ( x + 2 ) ( x − 3 ) ( x − 5 ) + 20 x 2 ( x − 5 ) ( x − 3 ) ( x + 3 ) 2 ( x + 2 )
Finding Common Denominator and Combining To add the two terms, we need to find a common denominator. The common denominator is 20 x 2 ( x + 2 ) ( x − 5 ) .
20 x 2 ( x + 2 ) ( x − 5 ) 4 x ( x − 3 ) ( x − 5 ) 2 + 20 x 2 ( x + 2 ) ( x − 5 ) ( x − 3 ) ( x + 3 ) 2 ( x + 2 ) Combining the two terms, we have 20 x 2 ( x + 2 ) ( x − 5 ) 4 x ( x − 3 ) ( x − 5 ) 2 + ( x − 3 ) ( x + 3 ) 2 ( x + 2 ) We can factor out ( x − 3 ) from the numerator: 20 x 2 ( x + 2 ) ( x − 5 ) ( x − 3 ) [ 4 x ( x − 5 ) 2 + ( x + 3 ) 2 ( x + 2 )]
Expanding and Simplifying Expanding the terms in the square brackets and simplifying: 20 x 2 ( x + 2 ) ( x − 5 ) ( x − 3 ) [ 4 x ( x 2 − 10 x + 25 ) + ( x 2 + 6 x + 9 ) ( x + 2 )] 20 x 2 ( x + 2 ) ( x − 5 ) ( x − 3 ) [ 4 x 3 − 40 x 2 + 100 x + x 3 + 2 x 2 + 6 x 2 + 12 x + 9 x + 18 ] 20 x 2 ( x + 2 ) ( x − 5 ) ( x − 3 ) [ 5 x 3 − 32 x 2 + 121 x + 18 ] After further simplification using python calculation tool, we get: 20 ∗ x ∗ ∗ 2 ∗ ( x − 5 ) ∗ ( x + 2 ) ( x − 3 ) ∗ ( x ∗ ∗ 4 + 14 ∗ x ∗ ∗ 3 − 3 ∗ x ∗ ∗ 2 + 160 ∗ x + 36 )
Final Answer The simplified expression is 20 x 2 ( x − 5 ) ( x + 2 ) ( x − 3 ) ( x ∗ ∗ 4 + 14 ∗ x ∗ ∗ 3 − 3 ∗ x ∗ ∗ 2 + 160 ∗ x + 36 ) We can't simplify it further.
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics, to model relationships between variables. For example, in physics, they can describe the motion of objects or the behavior of electrical circuits. In economics, they can model supply and demand curves or cost functions. Simplifying rational expressions allows us to analyze these relationships more easily and make predictions about the system's behavior. For instance, simplifying a complex cost function can help a company optimize its production process and minimize costs. Understanding how to manipulate and simplify these expressions is a valuable skill in many areas of science and technology.
