Factor each polynomial completely. See Example 6.t⁴ - 1

Verified step by step guidance

1. Recognize that the polynomial \(t^{4} - 1\) is a difference of squares because it can be written as \((t^{2})^{2} - 1^{2}\).
2. Apply the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\), so rewrite \(t^{4} - 1\) as \((t^{2} - 1)(t^{2} + 1)\).
3. Notice that \(t^{2} - 1\) is itself a difference of squares and can be factored further as \((t - 1)(t + 1)\).
4. The term \(t^{2} + 1\) is a sum of squares, which does not factor over the real numbers, so leave it as is.
5. Combine all factors to express the complete factorization: \((t - 1)(t + 1)(t^{2} + 1)\).

Key Concepts

Difference of Squares

The difference of squares is a factoring technique used when an expression is in the form a² - b². It factors into (a - b)(a + b). For example, t⁴ - 1 can be seen as (t²)² - 1², allowing it to be factored using this method.

Factoring Higher Powers

Polynomials with higher powers, like t⁴, can often be factored by recognizing patterns such as squares of squares or sums/differences of powers. Breaking down t⁴ - 1 into (t² - 1)(t² + 1) simplifies the expression further.

Factoring Quadratic Expressions

After applying difference of squares, resulting quadratic expressions like t² - 1 can be factored further if they are also differences of squares. This stepwise factoring continues until the polynomial is completely factored into irreducible factors.