Problem 39 - College Algebra Essentials

Question

Find each product. $$\left(1-y^{5}\right)\left(1+y^{5}\right)$$

Step-by-Step Solution

Answer

The product of \((1-y^{5})(1+y^{5})\) is \(1 - y^{10}\).

Step 1: Expand the Product

Start by expanding the product by distributing each term of the first binomial with each term of the second binomial. Here, \( (1*y^{5}) + (1*1) - (y^{5}*y^{5}) - (y^{5}*1) \). Note how, because of the different signs in the binomials, one subtraction is used instead of addition when multiplying \( y^{5}*y^{5} \).

Step 2: Simplify the Result

Simplify the above expression. You get \( y^{5} + 1 - y^{10} - y^{5} \). Notice how the \( y^{5} \) terms cancel out.

Step 3: Final Simplification

After the cancellation, the final simplified result is: \( 1 - y^{10} \). This is as simple as the expression can get, and there are no two terms that can be combined.