Problem 36 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The tenth term of \((x-1)^{12}\)

Step-by-Step Solution

Answer

The tenth term is \(-220x^3\).

Step 1: Understanding the Binomial Theorem

The Binomial Theorem states that \[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k.\]This helps us find any specific term in a binomial expansion without fully expanding it.

Step 2: Identifying the Variables

For the binomial \((x-1)^{12}\), we have:- \(a = x\)- \(b = -1\)- \(n = 12\).We want the 10th term, so we need the term where \(k=9\).

Step 3: Applying the Binomial Formula

The general formula for the \((k+1)\)-th term of a binomial expansion \((a+b)^n\) is:\[T_{k+1} = \binom{n}{k}a^{n-k}b^k.\]

Step 4: Calculating the Tenth Term

Substitute \(n = 12\), \(k = 9\), \(a = x\), and \(b = -1\) into the binomial formula:\[T_{10} = \binom{12}{9}x^{12-9}(-1)^9.\]

Step 5: Simplifying the Expression

Calculate \(\binom{12}{9}\) and evaluate powers:- \(\binom{12}{9} = \binom{12}{3} = 220\)- \(x^{3}\)- \((-1)^9 = -1\).Thus, \(T_{10} = 220x^3(-1) = -220x^3.\)

Key Concepts

Binomial Expansion

The binomial expansion is a powerful method used to express the power of a binomial expression like \((a + b)^n\) as a sum of terms. Each term in the expansion can be found systematically without having to multiply the entire expression. This is particularly useful when you only need a specific term.When you expand a binomial raised to a power, you're essentially applying the binomial theorem. This theorem provides a simple and direct way to find specific terms through the formula:\[(a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k.\] Here, the terms consist of a combination of the coefficients (which we'll discuss next), and the variables \(a\) and \(b\) raised to varying powers. This expansion is handy for problems where full expansion would be too tedious, such as calculating only one or a few specific terms.

Combinatorial Coefficients

Combinatorial coefficients, often represented as \(\binom{n}{k}\), are critical in calculating the terms of a binomial expansion. These coefficients are also known as binomial coefficients. They give you the number of ways to choose \(k\) items from \(n\) items without regard to the order.Understanding how to find these coefficients is important in simplifying the process of binomial expansion. The formula to calculate any binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!},\] where \(n!\) (read as \(n\) factorial) is the product of all positive integers up to \(n\).In our specific exercise, to find the tenth term of \((x-1)^{12}\), the coefficient involves \(\binom{12}{9}\). Through calculation, we know that:\[\binom{12}{9} = \binom{12}{3} = 220.\] This coefficient tells us that in the tenth term of our expansion, the numerical multiplier is 220 before it gets influenced by powers of \(x\) and constants like \(-1\).

Polynomial Expressions

Polynomial expressions are mathematical expressions that consist of variables and constants, combined using addition, subtraction, multiplication, and non-negative integer exponents. In the case of binomial expansions, we deal with polynomial expressions that have precisely two terms.Given the binomial \((x - 1)^{12}\), we need to find just one particular term in its expanded polynomial form without expanding the entire expression. The importance of understanding polynomial expressions in this context is realizing that each term in the expansion is itself a polynomial term, consisting of a variable part and a constant part.For the tenth term, by substituting into the formula \[T_{k+1} = \binom{n}{k} a^{n-k} b^k\],we calculated:

• The constant part: involves \(\binom{12}{9} = 220\) and \((-1)^9 = -1\).
• The variable part: involves \(x^{12-9} = x^3\).

Thus, the entire tenth term as a polynomial term becomes \(-220x^3\). Understanding these components is key to mastering the binomial expansion in polynomial form.