Problem 18 - Introductory Algebra for College Students

Question

Multiply as indicated. $$\left(y^{2}-16\right) \cdot \frac{3}{y-4}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(3(y + 4)\)

Step 1: Identify and factorize perfect squares

In this expression, \(y^{2}-16\) is a difference of squares as 16 is a perfect square. Difference of squares can be factorized as \((a+b)(a-b)\). So, \(y^{2} - 16\) can be factorized as \((y - 4)(y + 4)\).

Step 2: Replace the original expression

Replace the expression \(y^{2}-16\) in the original expression with its factorized form which gives \((y - 4)(y + 4) \cdot \frac{3}{y - 4}\)

Step 3: Simplify the expression

Here, we can directly cancel the same term \((y - 4)\) in the numerator and the denominator which results in \((y + 4) \cdot 3\).

Step 4: Write the final result

Rearrange the expression to get the final form, which is \(3(y + 4)\).