Find each product. See Example 5. (2x + 5)³

Verified step by step guidance

1. Recognize that the expression \((2x + 5)^3\) is a binomial raised to the third power, which means you need to expand it using the binomial cube formula.
2. Recall the binomial cube formula: \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). Here, identify \(a = 2x\) and \(b = 5\).
3. Calculate each term separately: \(a^3 = (2x)^3\), \(3a^2b = 3 \times (2x)^2 \times 5\), \(3ab^2 = 3 \times (2x) \times 5^2\), and \(b^3 = 5^3\).
4. Write the expanded form by summing all the terms: \((2x)^3 + 3(2x)^2(5) + 3(2x)(5)^2 + 5^3\).
5. Simplify each term by performing the powers and multiplications, then combine all terms to get the final expanded polynomial.

Key Concepts

Binomial Expansion

Binomial expansion is the process of expanding expressions raised to a power, such as (a + b)³. It involves applying the binomial theorem or using formulas to express the power as a sum of terms involving powers of each component.

Binomial Theorem

The binomial theorem provides a formula to expand expressions of the form (a + b)^n. It uses binomial coefficients, often represented by combinations, to determine the coefficients of each term in the expanded form.

Algebraic Multiplication of Polynomials

Multiplying polynomials involves distributing each term in one polynomial to every term in the other. For powers like cubes, this means repeated multiplication and combining like terms to simplify the expression.