Problem 64 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. $$\left(4 h-k^{4}\right)^{12} ; \text { last term }$$

Step-by-Step Solution

Answer

The last term of the binomial expansion \((4h-k^4)^{12}\) is \(k^{48}\).

Step 1: Identify the values of n, a, and b in the binomial theorem

We are given the binomial expansion \((4h-k^4)^{12}\). Comparing with the binomial theorem, we can see that n=12, a=4h, and b=-k^4.

Step 2: Determine the term number corresponding to the last term

We know that the last term has the highest power of b (-k^4). In the binomial theorem, there are a total of n+1 terms. In our case, there are 12+1=13 terms. The last term (term number 13) corresponds to the term with the highest power of b (-k^4).

Step 3: Apply the binomial coefficient formula to find the last term

Using the binomial theorem, the general term is \(\binom{n}{k}a^{n-k}b^k\). We need to find the term number 13, so k=12. Now substitute the known values n=12, a=4h, b=-k^4, and k=12: Last term = \(\binom{12}{12}(4h)^{12-12}(-k^4)^{12}\)

Step 4: Simplify the last term

Now simplify the last term: Last term = \(\binom{12}{12}(4h)^{0}(-k^4)^{12}\) The binomial coefficient \(\binom{12}{12} = 1\), and any non-zero number raised to the power of 0 is equal to 1. So we have: Last term = \(1 * 1 * (-1)^{12}k^{48}\) Since \((-1)^{12}\) is equal to 1, the last term simplifies to: Last term = \(k^{48}\) So, the last term of the binomial expansion \((4h-k^4)^{12}\) is \(k^{48}\).

Key Concepts

Binomial Expansion

The binomial theorem is a fundamental concept in algebra that helps us expand expressions that involve powers of a binomial. A binomial is an expression containing two terms, like

• \((a + b)^n\)

where \(n\) is a non-negative integer. The theorem allows us to express the binomial raised to a power as a sum of terms. Each term is a product of a binomial coefficient, and one term from each of the binomial factors is raised to different powers that sum up to \(n\).The formula for the binomial expansion looks like this:

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

where \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from a set of \(n\) elements. Understanding the binomial theorem makes it easier to simplify expressions and solve algebra problems involving polynomial expansions.

Algebra Problem Solving

Solving algebraic problems often involves breaking down complex expressions into simpler parts. When dealing with the binomial theorem and its expansions, it's helpful to identify the key components:

• The power \(n\) of the binomial.
• The individual terms \(a\) and \(b\) in the binomial.

In the given problem, we look at the binomial \((4h - k^4)^{12}\). Here, understanding each component allows us to determine the specific term we wish to find – in this case, the last term of the expansion.We calculate how many terms there are by adding one to the power of the binomial. So, there are \(12 + 1 = 13\) terms in total. For algebra problem-solving with binomials, one often uses the binomial formula to identify an individual term based on its position in the expansion. This technique makes it much easier to work with large polynomials.

Polynomial Expressions

Polynomial expressions like the one seen in the binomial \((4h - k^4)^{12}\) are often extended into multiple terms. Each term has its powers of the variables involved. When working with such expressions, we need to note:

• The structure involves exponents and coefficients.
• Powers of each component determine the exponents in each term.

For example, in our binomial expansion, the largest component contributed by \(-k^4\) leads to the term \(k^{48}\), showing how exponents compound in larger expressions.This expression of a polynomial, especially in high powers, allows detailed forms of algebraic manipulation. Recognizing the form of terms simplifies understanding and solving algebraic expressions. Polynomials appear everywhere in mathematics and understanding them, as demonstrated through binomial expansions, provides essential tools for solving more advanced mathematical problems.