Problem 40 - College Algebra

Question

Find each product. $$\left(2-y^{5}\right)\left(2+y^{5}\right)$$

Step-by-Step Solution

Answer

The product of \( \left(2-y^{5}\right)\left(2+y^{5}\right) \) is \( 4 - y^{10} \).

Step 1: Identify The Difference of Squares

The given expression is \( \left(2-y^{5}\right)\left(2+y^{5}\right) \). This is of the form \( (a - b)(a + b) \), which is a pattern for the difference of squares.

Step 2: Use The Difference of Squares Formula

We can use the formula of difference of squares to multiply this expression. The formula is \( (a - b)(a + b) = a^{2} - b^{2} \). Replace \( a \) with \( 2 \) and \( b \) with \( y^{5} \) in this formula.

Step 3: Compute The Square of \( a \) And \( b \)

Calculate the square of \( a \) and \( b \). We get \( a^{2} = 2^{2} = 4 \) and \( b^{2} = (y^{5})^{2} = y^{10} \).

Step 4: Substitute The Values

Substitute the calculated squares of \( a \) and \( b \) into the formula. We get the expression: \( 4 - y^{10} \).