Problem 73 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(5 x+2)^{3}$$

Step-by-Step Solution

Answer

\((5x+2)^3 = 125x^3 + 150x^2 + 60x + 8\).

Step 1: Formula for Binomial Expansion

To begin solving the problem \((5x+2)^3\), we use the binomial theorem, which states: \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] For our problem, \(a = 5x\), \(b = 2\), and \(n = 3\).

Step 2: Compute Binomial Coefficients

The binomial coefficients for \((5x+2)^3\) can be calculated using the formula \(\binom{n}{k}\). - \( \binom{3}{0} = 1 \)- \( \binom{3}{1} = 3 \)- \( \binom{3}{2} = 3 \)- \( \binom{3}{3} = 1 \)

Step 3: Expand the Binomial

Apply the binomial theorem with the coefficients:\((5x+2)^3 = \binom{3}{0}(5x)^3(2)^0 + \binom{3}{1}(5x)^2(2)^1 + \binom{3}{2}(5x)^1(2)^2 + \binom{3}{3}(5x)^0(2)^3\).

Step 4: Evaluate Each Term

Calculate each term in the expansion:- First term: \(1 \times (5x)^3 \times 1 = 125x^3\)- Second term: \(3 \times (5x)^2 \times 2 = 150x^2\)- Third term: \(3 \times (5x)^1 \times 4 = 60x\)- Fourth term: \(1 \times 1 \times 8 = 8\)

Step 5: Combine the Terms

Add all the terms obtained:\((5x+2)^3 = 125x^3 + 150x^2 + 60x + 8\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical formula used for expanding expressions that are raised to any power. Given a binomial expression in the form of \((a + b)^n\), the Binomial Theorem allows us to express it as a sum of terms. Each term reflects the original binomial raised to respective powers of its components, multiplied by specific coefficients.

The theorem is represented as:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

To understand it, consider that \(n\) denotes the power to which the binomial is raised. The expression is broken down into \(n+1\) terms, where each part contains a binomial coefficient \(\binom{n}{k}\), signifying the number of ways to choose \(k\) elements from \(n\).

This theorem is especially useful for simplifying computations, making it easier to expand polynomial expressions without repeatedly performing multiplication.

Binomial Coefficients

Binomial coefficients are key components in the Binomial Theorem, guiding us to correctly distribute the powers across the terms in a binomial expansion. They appear in triangular arrays known as Pascal's Triangle, and they indicate the number of ways to choose a set of objects from a larger set.

For a given expression like \((a+b)^n\), binomial coefficients are represented as \(\binom{n}{k}\), calculated with the formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Here, \(n!\) is the factorial of \(n\), which means the product of all positive integers up to \(n\). As a simple application, consider raising a binomial \((5x+2)^3\), where we find coefficients for each term by substituting \(n=3\) and the corresponding \(k\) values.

The resulting coefficients \(1, 3, 3, 1\) are then used to expand the expression, shaping the terms based on these values, effectively guiding the polynomial expansion.

Polynomial Expansion

Polynomial expansion involves expressing a binomial expression like \((5x+2)^3\) as a sum of simpler monomials. When expanded through the Binomial Theorem, the original expression becomes a combination of terms, each having a distinct structure.

The process begins with using the identified binomial coefficients for each term, paired with powers of each component of the binomial. For \((5x+2)^3\), the expanded polynomial is derived term-by-term:

• First term: \(1 \cdot (5x)^3 \cdot 1 = 125x^3\)
• Second term: \(3 \cdot (5x)^2 \cdot 2 = 150x^2\)
• Third term: \(3 \cdot (5x)^1 \cdot 4 = 60x\)
• Fourth term: \(1 \cdot 1 \cdot 8 = 8\)

Each monomial is constructed by multiplying its parts, gradually building the expanded form. All terms are summed to give the polynomial \(125x^3 + 150x^2 + 60x + 8\), providing a simplified representation of the initial binomial.