Problem 4 - Algebra for College Students

Question

\(f(x)=(x+3)^{4}+1\)

Step-by-Step Solution

Answer

\(x^4 + 12x^3 + 54x^2 + 108x + 82\).

Step 1: Expand using the Binomial Theorem

The function is given as \((x+3)^4 + 1\). To expand \((x+3)^4\), we use the Binomial Theorem, which states \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = x\), \(b = 3\), and \(n = 4\). The expansion is: \((x+3)^4 = \binom{4}{0}x^4 + \binom{4}{1}x^3 \cdot 3 + \binom{4}{2}x^2 \cdot 3^2 + \binom{4}{3}x \cdot 3^3 + \binom{4}{4} \cdot 3^4\).

Step 2: Calculate Binomial Coefficients

Compute the binomial coefficients: - \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\).

Step 3: Substitute and Simplify Terms

Substitute the binomial coefficients into the expanded form:\((x+3)^4 = 1\cdot x^4 + 4\cdot x^3 \cdot 3 + 6\cdot x^2 \cdot 9 + 4\cdot x \cdot 27 + 1\cdot 81\).Simplify each term:- \(x^4\)- \(12x^3\)- \(54x^2\)- \(108x\)- \(81\).

Step 4: Add the Constant Term

Combine the simplified polynomial with the constant term from the original function: \[(x+3)^4 + 1 = x^4 + 12x^3 + 54x^2 + 108x + 81 + 1\].Finally, simplify this to: \[x^4 + 12x^3 + 54x^2 + 108x + 82\].

Key Concepts

Polynomial Expansion

Polynomial expansion allows us to expand expressions that have power terms, like \((x+3)^4\), into a polynomial with multiple terms. The Binomial Theorem is used for this purpose. It's a fundamental tool for dealing with expressions in the form of \((a+b)^n\). By using the formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), we can expand the polynomial. The theorem lays out how to take a two-term expression raised to a power and break it down into a sum of terms, each consisting of powers of its constituent parts. This process makes it much easier to work with the expression in a useful form.

Binomial Coefficients

Binomial coefficients play a vital role in the expansion of binomials. They are represented as \(\binom{n}{k}\) and are calculated as \(\frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial. These coefficients help determine the weight of each term in the expansion of a binomial expression.

• When \( n = 4 \) and \( k \) varies from 0 to 4, we compute the coefficients: \(\binom{4}{0}=1\), \(\binom{4}{1}=4\), \(\binom{4}{2}=6\), \(\binom{4}{3}=4\), and \(\binom{4}{4}=1\).

These coefficients multiply each term in the expanded polynomial. Their role is to balance out the coefficients in the resulting polynomial, accounting for all possible combinations of powers of \(a\) and \(b\) in the expansion process.

Simplifying Expressions

Simplifying expressions is the final step in making polynomial expansions manageable and easy to interpret. Once we've expanded a polynomial using binomial coefficients, proper simplification involves combining like terms and carrying out any arithmetic needed to reach the simplest form. For our expression, \((x+3)^4 + 1\), simplifying involves calculating the numeric values in each term resulting from the expansion. Each term resulting from the application of the binomial coefficients is simplified:

• \(1 \cdot x^4 = x^4\)
• \(4 \cdot x^3 \cdot 3 = 12x^3\)
• \(6 \cdot x^2 \cdot 9 = 54x^2\)
• \(4 \cdot x \cdot 27 = 108x\)
• \(1 \cdot 81 = 81\)

After the expansion and simplification, we also incorporate any constant terms from the original function, merging all components into their simplest expression, yielding \(x^4 + 12x^3 + 54x^2 + 108x + 82\).