Problem 52 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). \((a+b)^{1}=a+b\) \((a+b)^{2}=a^{2}+2 a b+b^{2}\) \((a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}\) \((a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4}\) \((a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\) Describe the pattern for the sum of the exponents on the variables in each term.

Step-by-Step Solution

Answer

In the expanded form of the binomial expression \((a+b)^{n}\), the sum of the exponents on a and b in each term is always equal to n, the original exponent in the binomial expression. This pattern holds true for all the binomial expressions provided in the exercise.

Step 1: Observe the pattern

Looking at the expanded form of \((a+b)^{n}\), we can see that each term consists of the variables a and b raised to some powers. Let's take \((a+b)^{5}\) as an example: \(a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\). In this expanded form, the exponents on the variables a and b change in each term.

Step 2: Examine the exponents

In each term of the expansion, the exponents on a decrease by 1 in every next term, starting from 5. Concurrently, the exponents on b increase by 1 in every next term, starting from 0. For instance, in the first term, \(a^{5}\), the exponent of a is 5 and b is 0; in the next term, \(5a^{4}b\), the exponent of a has decreased by 1 to become 4, and the exponent of b has increased by 1 to become 1.

Step 3: Identify the pattern

Taking a closer look, we can see that the sum of the exponents on a and b in each term is always equal to the original power n. For instance, in the term \(10 a^{2} b^{3}\), the sum of the exponents 2 and 3 equals 5, which was the original exponent. This pattern can be confirmed by reviewing the terms in other binomial expansions as well.

Key Concepts

Exponents

Exponents are a fundamental concept in algebra that significantly simplify the process of repeated multiplication. When you see an expression like \(a^n\), the exponent \(n\) tells you how many times to multiply the base \(a\) by itself. For example, \(a^3 = a \times a \times a\). In the context of binomial expansions such as \((a + b)^n\), exponents are used to express the repeated application of the binomial term, simplifying the expression into a polynomial form.

In a binomial expansion, exponents change in a predictable pattern. For any given term in the expansion, the exponents of one of the variables decrease progressively, while the exponents of the other variable increase correspondingly. This concept is crucial to understanding the arrangement of each term in the expansion. For example, in \((a+b)^5\), the first term is \(a^5\) and the last is \(b^5\), with each subsequent term adjusting the exponents while keeping their total equal to 5, the original exponent value.

Binomial Theorem

The Binomial Theorem provides a powerful method for expanding binomials raised to any given power \(n\). It states that \((a+b)^n\) can be expanded into a sum involving terms of the form \(\binom{n}{k} a^{n-k}b^k\), where \(\binom{n}{k}\) is a binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).

This theorem is useful for quickly finding each term in the expansion without having to manually multiply out every factor. The binomial coefficients mark the number of ways to choose \(k\) items from \(n\), creating the symmetrical pattern of coefficients in the expanded form. For instance, in \((a+b)^3\), the coefficients of the terms are 1, 3, 3, and 1, calculated using the theorem as \(\binom{3}{0}\), \(\binom{3}{1}\), \(\binom{3}{2}\), and \(\binom{3}{3}\).
The sum of the exponents in each term equals \(n\), creating the balanced polynomial expression.

Polynomial Expansion

Polynomial expansion involves expressing a compact expression, like a term raised to an exponent, as a sum of multiple terms. In the context of the binomial theorem, this involves expanding \((a+b)^n\) into a series of terms where both \(a\) and \(b\) are raised to various powers.

Each term in the polynomial expansion has its unique structure, combining the bases with specific coefficients and exponents. The exponents of \(a\) and \(b\) in each term of the expansion always add up to \(n\), the exponent in the original binomial. For example, in the expansion of \((a+b)^4\), we have terms like \(a^4\), \(4a^3b\), \(6a^2b^2\), \(4ab^3\), and \(b^4\), all governed by this balancing rule.

• This pattern and structure reveals insights into the underlying symmetry and combinatorial properties of polynomials.
• Each term is carefully structured to reflect the dynamics of repeated multiplication as specified by the binomial theorem.

Polynomial expansions are incredibly useful for simplifying complex expressions and solving numerous algebraic problems efficiently.