Problem 80 - Introductory Algebra for College Students

Question

Simplify each rational expression. $$\frac{16-y^{2}}{y(y-8)+16}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \(-\frac{(y+4)}{y-4}\).

Step 1: Recognize the Difference of Squares on the Numerator

The numerator, \(16-y^{2}\), is a difference of squares. This type of expression can be factored into the product of the sums and differences of the two terms i.e., \((a^{2}-b^{2}) = (a+b)(a-b)\). So, \(16-y^{2}\) can be factored as \((4+y)(4-y)\).

Step 2: Simplify the Denominator

The denominator, \(y(y-8)+16\), can be simplified by factoring. Distribute y into the brackets, i.e, \(y^2 - 8y + 16\). Notice that this can be factored into a perfect square \((y-4)^2\).

Step 3: Simplify the Rational Expression

Now, let's simplify the rational expression: \(\frac{(4+y)(4-y)}{(y-4)^{2}}\). Note that \((4-y)\) and \((y-4)\) are the same, just that one is negative of the other. One \((y-4)\) from the denominator cancels out one of either \((4-y)\) or \((4+y)\) from the numerator, but this cancelation would carry a minus sign. Therefore, you get: \(-\frac{(4+y)}{(y-4)}\).

Step 4: Final Simplification

Note that both the numerator and denominator have common factors, which causes confusion sometimes. So, let's rewrite \(4+y\) as \(y + 4\), yielding a final simplified expression of \(-\frac{(y+4)}{(y-4)}\).