Questions: Hence express ((2/(3-x))+(3/(1+x))) * ((x^2+8x-33)/(121-x^2)) as a single fraction in its lowest terms.

Solution Steps

To express the given expression as a single fraction in its lowest terms, we need to follow these steps:

1. Find a common denominator for the two fractions inside the parentheses.
2. Combine the fractions into a single fraction using the common denominator.
3. Multiply the resulting fraction by the third fraction.
4. Simplify the resulting expression by factoring and canceling common terms.

We start with the expression

\[ \left(\frac{2}{3-x} + \frac{3}{1+x}\right) \times \frac{x^{2}+8x-33}{121-x^{2}}. \]

To combine the fractions inside the parentheses, we find a common denominator, which is \((3-x)(1+x)\). Thus, we rewrite the expression as:

\[ \frac{2(1+x) + 3(3-x)}{(3-x)(1+x)}. \]

Next, we simplify the numerator:

\[ 2(1+x) + 3(3-x) = 2 + 2x + 9 - 3x = 11 - x. \]

So, the combined fraction becomes:

\[ \frac{11 - x}{(3-x)(1+x)}. \]

Now we multiply this result by the third fraction:

\[ \frac{11 - x}{(3-x)(1+x)} \times \frac{x^{2}+8x-33}{121-x^{2}}. \]

The expression simplifies to:

\[ \frac{(11 - x)(x^{2}+8x-33)}{(3-x)(1+x)(121-x^{2})}. \]

After simplification, we find that the expression reduces to:

\[ -\frac{1}{1+x}. \]

Final Answer