Subtract. Simplify the result if possible.
[tex]$\frac{5 x^2+8 x}{x-4}-\frac{27 x+4}{x-4}$[/tex]
[tex]$\frac{5 x^2+8 x}{x-4}-\frac{27 x+4}{x-4}= \square$[/tex] (Simplify your answer.)
Answer (1)
Combine the numerators over the common denominator: x − 4 ( 5 x 2 + 8 x ) − ( 27 x + 4 ) .
Simplify the numerator: x − 4 5 x 2 − 19 x − 4 .
Factor the numerator: x − 4 ( 5 x + 1 ) ( x − 4 ) .
Cancel the common factor: 5 x + 1 . The final answer is 5 x + 1 .
Explanation
Understanding the Problem We are asked to subtract two rational expressions with the same denominator. This makes the subtraction straightforward, as we can combine the numerators over the common denominator.
Restating the Expression The given expression is: x − 4 5 x 2 + 8 x − x − 4 27 x + 4
Combining Numerators Since the denominators are the same, we can combine the numerators: x − 4 ( 5 x 2 + 8 x ) − ( 27 x + 4 )
Simplifying the Numerator Now, we simplify the numerator by distributing the negative sign and combining like terms: x − 4 5 x 2 + 8 x − 27 x − 4 = x − 4 5 x 2 − 19 x − 4
Factoring the Numerator Next, we try to factor the numerator to see if we can simplify the rational expression further. We are looking for two numbers that multiply to 5 × − 4 = − 20 and add up to − 19 . These numbers are − 20 and 1 . So we can rewrite the middle term as − 20 x + x : 5 x 2 − 19 x − 4 = 5 x 2 − 20 x + x − 4 Now, we factor by grouping: 5 x 2 − 20 x + x − 4 = 5 x ( x − 4 ) + 1 ( x − 4 ) = ( 5 x + 1 ) ( x − 4 )
Simplifying the Expression Now we have: x − 4 ( 5 x + 1 ) ( x − 4 ) We can cancel the common factor of ( x − 4 ) from the numerator and the denominator, provided that x = 4 : x − 4 ( 5 x + 1 ) ( x − 4 ) = 5 x + 1
Final Answer Therefore, the simplified expression is 5 x + 1 .
Examples
Rational expressions are useful in various fields, such as physics and engineering, for modeling relationships between different quantities. For example, in circuit analysis, the impedance of a circuit can be represented as a rational expression involving the frequency of the input signal. Simplifying these expressions can help engineers understand the behavior of the circuit and design it effectively. Also, in economics, rational functions can model cost-benefit ratios, where simplifying these functions can provide insights into optimizing resource allocation and maximizing profits. Understanding how to manipulate and simplify rational expressions is therefore a valuable skill in many practical applications.
