Problem 48 - A Graphical Approach to College Algebra

Question

Find the term in the expansion of \((3+\sqrt{x})^{11}\) that contains \(x^{4}\).

Step-by-Step Solution

Answer

The term is \(4455x^4\).

Step 1: Understanding the Problem

We need to find the term in the expansion of \((3+\sqrt{x})^{11}\) that contains \(x^{4}\). We will use the binomial theorem here.

Step 2: Using the Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Substituting \(a = 3\), \(b = \sqrt{x}\), and \(n = 11\) into the formula will help us find the term involving \(x^4\).

Step 3: Determining the Relevant Term

We want to find the term where \(b^k = (\sqrt{x})^k = x^{4}\). This implies that \(k = 8\), since \((\sqrt{x})^8 = x^4\).

Step 4: Substitute Back into the Formula

Using \(k = 8\), substitute into the binomial formula to find the specific term: \[ \binom{11}{8} 3^{11-8} (\sqrt{x})^8 \].

Step 5: Simplifying the Expression

Calculate \(\binom{11}{8} = \binom{11}{3} = 165\) and \(3^3 = 27\). Substituting these into the expression gives us \(165 \times 27 \times x^4\).

Step 6: Final Calculation

The term simplifies to \(4455x^4\). Hence, the required term in the expansion is \(4455x^4\).

Key Concepts

Polynomial Expansion

When working with polynomials, one of the key techniques to understand is expansion. Polynomial expansion involves expressing a polynomial, typically written as a power or a product, in an expanded form. This process often utilizes the binomial theorem, especially when dealing with expressions like \[(a + b)^n\].By using the binomial theorem, you can expand the expression into a sum of terms, where each term involves a combination of the original variables raised to certain powers. Each term in the expanded form corresponds to a specific balance between the two original terms \(a\) and \(b\).

• The coefficients of these terms follow a distinct pattern known as binomial coefficients, which can be found in Pascal's Triangle.
• Each term of the expansion is formed by multiplying the binomial coefficient with the appropriate powers of \(a\) and \(b\).
• This method is especially useful for expanding expressions raised to high powers, where manual multiplication would be time-consuming and prone to error.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. In solving problems like the polynomial expansion of \((3+\sqrt{x})^{11}\), algebra comes into play significantly.The steps we take involve adjusting variables and expressions, making algebra a foundational tool. Here, the algebraic manipulation includes not just the use of the binomial theorem, but also substitution and simplification.

• Understanding algebra helps in determining what operations can simplify or solve a problem.
• In our given exercise, identifying and manipulating the right exponents and coefficients required an algebraic approach.
• Algebra helps in understanding the relationships and patterns within mathematical expressions.

Mastery of basic algebra is crucial for interpreting and solving complex polynomial expressions effectively.

Exponents

In mathematics, exponents are used to represent repeated multiplication. Calculating powers and manipulating them can be complex, particularly when dealing with composite expressions like \( (3 + \sqrt{x})^{11} \).Understanding how exponents work is vital when applying the binomial theorem. Here, exponents appear in several places:

• The expression involves raising a binomial to the eleventh power \((11)\), a direct use of exponents.
• In our example, recognizing that \( (\sqrt{x})^8 \) equals \(x^4\) was key to identifying the specific term of interest. This involves understanding how a square root can be transformed using exponents.
• Exponents also appear in the coefficients calculation, as we computed \(3^3\), which is an exponentiation.

Mastering the manipulation of exponents simplifies solving polynomial expansions and other algebraic problems. Being comfortable with handling powers, roots, and exponent rules is essential for tackling advanced math challenges.