Select the correct answer.
What is the simplest form of this expression?
[tex]$\frac{x-3}{x-1}+\frac{6}{x-3}$[/tex]
A. [tex]$\frac{x+3}{(x-1)(x-3)}$[/tex]
B. [tex]$\frac{x^2-6 x+3}{(x-1)(x-3)}$[/tex]
C. [tex]$\frac{x^2+3}{(x-1)(x-3)}$[/tex]
D. [tex]$\frac{x^2+12 x+15}{(x-1)(x-3)}$[/tex]
Answer (1)
Find a common denominator: ( x − 1 ) ( x − 3 ) .
Rewrite the fractions with the common denominator and combine the numerators: ( x − 1 ) ( x − 3 ) ( x − 3 ) ( x − 3 ) + 6 ( x − 1 ) .
Expand and simplify the numerator: ( x − 1 ) ( x − 3 ) x 2 − 6 x + 9 + 6 x − 6 = ( x − 1 ) ( x − 3 ) x 2 + 3 .
The simplest form of the expression is ( x − 1 ) ( x − 3 ) x 2 + 3 .
Explanation
Understanding the Problem We are given the expression x − 1 x − 3 + x − 3 6 . Our goal is to simplify this expression and identify the correct simplified form from the provided options.
Finding a Common Denominator To simplify the expression, we need to find a common denominator for the two fractions. The common denominator is ( x − 1 ) ( x − 3 ) . We rewrite each fraction with this common denominator: x − 1 x − 3 + x − 3 6 = ( x − 1 ) ( x − 3 ) ( x − 3 ) ( x − 3 ) + ( x − 1 ) ( x − 3 ) 6 ( x − 1 )
Combining the Fractions Now, we combine the numerators over the common denominator: ( x − 1 ) ( x − 3 ) ( x − 3 ) ( x − 3 ) + 6 ( x − 1 ) Next, we expand the numerator: ( x − 1 ) ( x − 3 ) x 2 − 6 x + 9 + 6 x − 6
Simplifying the Expression We simplify the numerator by combining like terms: ( x − 1 ) ( x − 3 ) x 2 + 3
Identifying the Correct Option Finally, we compare the simplified expression with the given options. The simplified expression is ( x − 1 ) ( x − 3 ) x 2 + 3 , which matches option C.
Examples
Simplifying rational expressions is a fundamental skill in algebra, with applications in various fields. For instance, in physics, you might encounter such expressions when dealing with electrical circuits or fluid dynamics. Imagine you're designing a bridge and need to calculate the stress distribution. Simplifying complex rational expressions helps in making these calculations more manageable and accurate, ensuring the bridge's structural integrity.
