In Exercises 15–58, find each product. (1−y^5)(1+y^5)

Verified step by step guidance

1. Recognize that the expression $(1-y^5)(1+y^5)$ is a difference of squares.
2. Recall the formula for the difference of squares: $(a-b)(a+b)=a^2-b^2$.
3. Identify $a=1$ and $b=y^5$ in the expression.
4. Apply the difference of squares formula: $(1{)}^2-(y^5{)}^2$.
5. Simplify the expression to get $1-y^{10}$.

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that states that for any two terms a and b, the expression (a - b)(a + b) equals a² - b². This identity is crucial for simplifying expressions that involve the product of a sum and a difference, allowing for easier calculations and factorizations.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can simplify expressions and solve equations. Understanding how to factor polynomials, especially using identities like the difference of squares, is essential for manipulating algebraic expressions effectively.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the expression (1 - y^5)(1 + y^5), recognizing that y^5 is a power allows for the application of the difference of squares identity. Mastery of exponents is vital for simplifying expressions and solving equations in algebra.