Factor each polynomial. See Examples 5 and 6. 25s4-9t2

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 63

1장, 문제 63

Factor each polynomial. See Examples 5 and 6. 25s⁴-9t²

검증된 단계별 안내

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

Recall the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = 5s^2\) and \(b = 3t\) in this case.

Apply the difference of squares formula to factor the polynomial as \((5s^2 - 3t)(5s^2 + 3t)\).

Check each factor to see if it can be factored further. Notice that \(5s^2 - 3t\) and \(5s^2 + 3t\) are not difference or sum of squares and cannot be factored further over the real numbers.

Write the final factored form as \((5s^2 - 3t)(5s^2 + 3t)\).

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주요 개념

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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.

Factoring Higher Powers

When variables are raised to powers 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.

Prime Factorization of Coefficients

Breaking down numerical coefficients into their prime factors helps identify perfect squares and simplifies the factoring process. For example, 25 is 5² and 9 is 3², which aids in recognizing the difference of squares structure.