Problem 20 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$(2 a+b)^{3}$$

Step-by-Step Solution

Answer

The expression \((2a + b)^3\) is \(8a^3 + 12a^2b + 6ab^2 + b^3\).

Step 1: Understand the Expression

We need to express \((2a + b)^3\) as a polynomial. This expression indicates we are cubing the sum \(2a + b\).

Step 2: Use the Binomial Theorem

The binomial theorem states that \((x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). Here, \(x = 2a\), \(y = b\), and \(n = 3\).

Step 3: Calculate Each Term with Binomial Coefficients

We calculate each term for \(k = 0, 1, 2, 3\):- For \(k=0\): \(\binom{3}{0} (2a)^3 b^0 = 1 imes 8a^3 = 8a^3\)- For \(k=1\): \(\binom{3}{1} (2a)^2 b^1 = 3 imes 4a^2 b = 12a^2 b\)- For \(k=2\): \(\binom{3}{2} (2a)^1 b^2 = 3 imes 2a b^2 = 6a b^2\)- For \(k=3\): \(\binom{3}{3} (2a)^0 b^3 = 1 imes b^3 = b^3\)

Step 4: Combine All Terms

Now, sum all the terms obtained from the previous step: \(8a^3 + 12a^2b + 6ab^2 + b^3\).

Key Concepts

Polynomial Expansion

Polynomial expansion is a technique used in mathematical operations to express a power of a binomial as a sum of terms. Each term in this expansion is a monomial: a single term made of a constant called the coefficient, variables raised to power, and sometimes both. For example, when we expand (2a + b)^3, we transform it from a compact form into a longer polynomial with several terms.

• Each term in the expansion represents a part of the whole, making it easier to perform certain operations like addition or integration.
• A polynomial is expressed in the sum of multiple terms with varying combinations of the variables.

Using expansion allows you to systematically break down complex expressions into manageable pieces, illuminating properties such as symmetry and the individual influence of each variable.

Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in the expansion of a binomial raised to a power. These coefficients can be found using the famous Pascals Triangle or by employing the combination formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n \) is the power of the binomial and \( k \) is the term number in the sequence.

• For the expansion of (2a + b)^3, the coefficients are 1, 3, 3, and 1. These align with the fourth row of Pascal's Triangle, which correlates with a binomial expansion of degree 3.
• Each coefficient tells us how many specific terms of the form \(x^{n-k} \times y^k\) are present in the expansion.

The use of binomial coefficients enables precise calculation of polynomial terms, ensuring each expansion is accurate and complete.

Power of a Binomial

A binomial is an algebraic expression that has exactly two terms, like (x + y). Understanding the "power of a binomial" means recognizing how to expand this term when it is raised to a non-negative integer power. This expansion can be systematically achieved using the Binomial Theorem:
\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]Using this theorem, we can compute the polynomial form of any binomial raised to a power.

• For instance, with (2a + b)^3, the power indicates how many times you multiply the binomial by itself.
• Each term in the expanded polynomial embeds information from the multiplication process and the original binomial terms' relative contributions.

Raising a binomial to a power captures all possible products of its terms in a comprehensive polynomial.