In Exercises 15–58, find each product. (4x²+5x)(4x²−5x)

Verified step by step guidance

1. Recognize that the given expression is a product of two binomials: \((4x^2 + 5x)(4x^2 - 5x)\). This is a difference of squares pattern.
2. Recall the formula for the difference of squares: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 4x^2\) and \(b = 5x\).
3. Apply the formula: \((4x^2)^2 - (5x)^2\).
4. Simplify each term: \((4x^2)^2 = 16x^4\) and \((5x)^2 = 25x^2\).
5. Combine the results to write the simplified expression: \(16x^4 - 25x^2\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This process requires applying the distributive property, ensuring that all combinations of terms are accounted for. For example, in the expression (a + b)(c + d), you would calculate ac, ad, bc, and bd, then combine like terms.

Difference of Squares

The difference of squares is a specific algebraic identity that states a^2 - b^2 = (a + b)(a - b). This identity is useful for simplifying expressions where one polynomial is the square of a term and the other is the square of another term. In the given problem, recognizing that (4x^2 + 5x)(4x^2 - 5x) can be treated as a difference of squares can simplify the multiplication process.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is crucial after performing polynomial multiplication, as it helps to present the final answer in its simplest form. For instance, in the expression 3x^2 + 2x^2, you would combine the like terms to get 5x^2.