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.
2. Use the Binomial Theorem, which states $(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$, to expand the expression.
3. Identify $a=2x$, $b=5$, and $n=3$ in the binomial expression.
4. Calculate each term of the expansion: $(\genfrac{}{}{0ex}{}{3}{0})(2x{)}^3(5{)}^0$, $(\genfrac{}{}{0ex}{}{3}{1})(2x{)}^2(5{)}^1$, $(\genfrac{}{}{0ex}{}{3}{2})(2x{)}^1(5{)}^2$, $(\genfrac{}{}{0ex}{}{3}{3})(2x{)}^0(5{)}^3$.
5. Combine all the terms to get the expanded form of the expression.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions raised to a power, specifically for binomials of the form (a + b)ⁿ. It states that (a + b)ⁿ = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem is essential for efficiently calculating powers of binomials without multiplying them out directly.

Polynomial Expansion

Polynomial expansion involves rewriting a polynomial expression in a simplified form, often by distributing terms and combining like terms. In the case of (2x + 5)³, this requires applying the Binomial Theorem or using the distributive property multiple times to achieve the final expanded form. Understanding this process is crucial for solving polynomial equations and simplifying expressions.

Coefficients and Combinations

In the context of the Binomial Theorem, coefficients represent the numerical factors in front of the terms in the expansion. These coefficients can be determined using combinations, specifically 'n choose k', which calculates how many ways you can choose k elements from a set of n elements. Recognizing how to compute these coefficients is vital for accurately expanding binomials.