Find each product. See Example 5. (4x² - 5y) (4x² + 5y)

검증된 단계별 안내

Recognize that the expression is a product of two binomials in the form \((a - b)(a + b)\), where \(a = 4x^{2}\) and \(b = 5y\).

Recall the difference of squares formula: \((a - b)(a + b) = a^{2} - b^{2}\).

Apply the formula by squaring each term: calculate \(a^{2} = (4x^{2})^{2}\) and \(b^{2} = (5y)^{2}\).

Write the expression as \(a^{2} - b^{2}\), which becomes \((4x^{2})^{2} - (5y)^{2}\).

Simplify the squares by applying the exponent rules: \((4x^{2})^{2} = 4^{2} imes (x^{2})^{2}\) and \((5y)^{2} = 5^{2} imes y^{2}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Difference of Squares

The difference of squares is a special product formula: (a - b)(a + b) = a² - b². It simplifies the multiplication of two binomials that are conjugates by subtracting the square of the second term from the square of the first.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms. This process is essential for expanding expressions and simplifying products.

Exponents and Like Terms

Understanding exponents is crucial when squaring terms like 4x², which becomes (4x²)² = 16x⁴. Combining like terms means adding or subtracting terms with the same variable and exponent to simplify the final expression.