Simplify \( \frac{11 x}{x^{2}-6 x+9}+\frac{11}{x-3} \) and give the domain. \( x \neq \square \) State the sum in simplest form. \( \frac{11 x}{x^{2}-6 x+9}+\frac{11}{x-3}=\square \)
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To simplify the expression \( \frac{11 x}{x^{2}-6 x+9}+\frac{11}{x-3} \), we first note that \( x^2 - 6x + 9 \) factors into \( (x-3)^2 \). Thus, we rewrite the expression: \[ \frac{11 x}{(x-3)^2} + \frac{11}{x-3} \] Next, we need a common denominator, which is \( (x-3)^2 \): \[ \frac{11 x}{(x-3)^2} + \frac{11(x-3)}{(x-3)^2} \] Combining the fractions gives us: \[ \frac{11 x + 11(x-3)}{(x-3)^2} \] Simplifying the numerator: \[ 11 x + 11(x-3) = 11 x + 11 x - 33 = 22 x - 33 \] Thus, we have: \[ \frac{22 x - 33}{(x-3)^2} \] Factoring out a common factor of 11 from the numerator: \[ \frac{11(2x - 3)}{(x-3)^2} \] Now, we analyze the domain. The expression is undefined when the denominator is zero, i.e., when \( (x-3)^2 = 0 \), leading to \( x = 3 \). Thus, the domain is \( x \neq 3 \). The sum in simplest form is: \[ \frac{11(2x - 3)}{(x-3)^2} \] To summarize: The domain is \( x \neq 3 \). The simplest form is \( \frac{11(2x - 3)}{(x-3)^2} \).
