Factor each polynomial completely. See Example 6. 25s⁴ - 9t²

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Recognize that the given expression \(25s^{4} - 9t^{2}\) is a difference of squares because it can be written as \((5s^{2})^{2} - (3t)^{2}\).

Apply the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\), where \(a = 5s^{2}\) and \(b = 3t\).

Rewrite the expression as \((5s^{2} - 3t)(5s^{2} + 3t)\) after applying the difference of squares factorization.

Check if either factor can be factored further. Notice that \(5s^{2} - 3t\) and \(5s^{2} + 3t\) are not difference or sum of squares or any other common factorable forms.

Conclude that the complete factorization of the polynomial is \((5s^{2} - 3t)(5s^{2} + 3t)\).

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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). Recognizing this pattern helps simplify polynomials like 25s⁴ - 9t² by identifying perfect squares.

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Sum and Difference of Tangent

Factoring Higher Powers

When variables have exponents greater than 2, such as s⁴, it can be helpful to rewrite them as powers squared (e.g., s⁴ = (s²)²). This allows the use of difference of squares or other factoring methods on more complex terms.

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Factoring

Prime Factorization of Coefficients

Breaking down numerical coefficients into their prime factors helps identify perfect squares and simplifies the factoring process. For example, 25 and 9 are perfect squares (5² and 3²), which is essential for applying the difference of squares method.

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Percorso guidato

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Factoring

Pratica correlata

Domanda del libro di testo

Use an inequality symbol to write each statement. 5 is greater than or equal to 5.

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Domanda del libro di testo

Determine the intervals of the domain over which each function is continuous. See Example 9.

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