Factor each polynomial. See Examples 5 and 6. y⁴-81

Verified step by step guidance

1. Recognize that the polynomial \(y^4 - 81\) is a difference of squares because it can be written as \((y^2)^2 - 9^2\).
2. Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = y^2\) and \(b = 9\).
3. Rewrite the expression as \((y^2 - 9)(y^2 + 9)\) after factoring using the difference of squares.
4. Notice that \(y^2 - 9\) is itself a difference of squares and can be factored further as \((y - 3)(y + 3)\).
5. Write the fully factored form as \((y - 3)(y + 3)(y^2 + 9)\), where \(y^2 + 9\) cannot be factored further over the real numbers.

Key Concepts

Difference of Squares

The difference of squares is a factoring technique used when a polynomial is expressed as a subtraction between two perfect squares, such as a² - b². It factors into (a - b)(a + b). Recognizing this pattern simplifies factoring expressions like y⁴ - 81.

Factoring Higher Powers

Polynomials with higher powers, like y⁴, can often be rewritten as powers of squares, e.g., (y²)². This allows the use of difference of squares repeatedly to break down complex expressions into simpler factors.

Perfect Squares

A perfect square is a number or expression that can be written as the square of another number or expression, such as 81 = 9². Identifying perfect squares in a polynomial is essential for applying the difference of squares method effectively.