How do you find the term in an expansion?
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Learning Objectives
A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find \((x+y)^n\) without multiplying the binomial by itself \(n\) times. Identifying Binomial CoefficientsIn the shortcut to finding \({(x+y)}^n\), we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation \(\dbinom{n}{r}\) instead of \(C(n,r)\), but it can be calculated in the same way. So \[\dbinom{n}{r}=C(n,r)=\dfrac{n!}{r!(n−r)!}\] The combination \(\dbinom{n}{r}\) is called a binomial coefficient. An example of a binomial coefficient is: \(\dbinom{5}{2}=C(5,2)=10\) Definition: BINOMIAL COEFFICIENTS If \(n\) and \(r\) are integers greater than or equal to \(0\) with \(n≥r\), then the binomial coefficient is \[\dbinom{n}{r}=C(n,r)=\dfrac{n!}{r!(n−r)!} \label{binomial1}\] Q&A: Is a binomial coefficient always a whole number? Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number. Example \(\PageIndex{1}\): Finding Binomial Coefficients Find each binomial coefficient.
Solution Use the Equation \ref{binomial1} to calculate each binomial coefficient. You can also use the \(nC_r\) function on your calculator.
Analysis Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations. \[\dbinom{n}{r}=\dbinom{n}{n−r} \nonumber\] Exercise \(\PageIndex{1}\) Find each binomial coefficient.
\(35\) Answer b\(33\) Using the Binomial TheoremWhen we expand \({(x+y)}^n\) by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand \({(x+y)}^{52}\), we might multiply \((x+y)\) by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions. \[\begin{align*} {(x+y)}^2 &= x^2+2xy+y^2 \\[4pt] {(x+y)}^3 &= x^3+3x^2y+3xy^2+y^3 \\[4pt] {(x+y)}^4 &= x^4+4x^3y+6x^2y^2+4xy^3+y^4 \end{align*}\] First, let’s examine the exponents. With each successive term, the exponent for \(x\) decreases and the exponent for \(y\) increases. The sum of the two exponents is \(n\) for each term. Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern: \(\dbinom{n}{0}\), \(\dbinom{n}{1}\), \(\dbinom{n}{2}\),..., \(\dbinom{n}{n}.\) These patterns lead us to the Binomial Theorem, which can be used to expand any binomial. \[\begin{align*} {(x+y)}^n&=\sum_{k=0}^{n}\dbinom{n}{k}x^{n−k}y^k \\[4pt] &=x^n+\dbinom{n}{1}x^{n−1}y+\dbinom{n}{2}x^{n−2}y^2+...+\dbinom{n}{n−1}xy^{n−1}+y^n \end{align*}\] Another way to see the coefficients is to examine the expansion of a binomial in general form, \(x+y\), to successive powers \(1\), \(2\), \(3\), and \(4\). \[\begin{align*} {(x+y)}^1 &= x+y \\ {(x+y)}^2 &= x^2+2xy+y^2 \\ {(x+y)}^3 &= x^3+3x^2y+3xy^2+y^3 \\ {(x+y)}^4 &= x^4+4x^3y+6x^2y^2+4xy^3+y^4 \end{align*}\] Can you guess the next expansion for the binomial \({(x+y)}^5\)? Figure \(\PageIndex{1}\) See Figure \(\PageIndex{1}\), which illustrates the following:
To determine the expansion on \({(x+y)}^5\), we see \(n=5\), thus, there will be \(5+1=6\) terms. Each term has a combined degree of \(5\). In descending order for powers of \(x\), the pattern is as follows:
The next expansion would be \({(x+y)}^5=x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5\) But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal's Triangle, shown in Figure \(\PageIndex{2}\). Figure \(\PageIndex{2}\) To generate Pascal’s Triangle, we start by writing a \(1\). In the row below, row 2, we write two \(1’s\). In the 3rd row, flank the ends of the rows with \(1’s\), and add \(1+1\) to find the middle number, \(2\). In the \(n^{th}\) row, flank the ends of the row with \(1’s\). Each element in the triangle is the sum of the two elements immediately above it. To see the connection between Pascal’s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form. THE BINOMIAL THEOREM The Binomial Theorem is a formula that can be used to expand any binomial. \[ {(x+y)}^n = \sum_{k=0}^{n}\dbinom{n}{k}x^{n−k}y^k = x^n+\dbinom{n}{1}x^{n−1}y+\dbinom{n}{2}x^{n−2}y^2+...+\dbinom{n}{n−1}xy^{n−1}+y^n \] How to: Given a binomial, write it in expanded form.
Example \(\PageIndex{2}\): Expanding a Binomial Write in expanded form.
Solution a. Substitute \(n=5\) into the formula. Evaluate the \(k=0\) through \(k=5\) terms. Simplify. \[\begin{align*} {(x+y)}^5 &= \dbinom{5}{0}x^5y^0+\dbinom{5}{1}x^4y^1+\dbinom{5}{2}x^3y^2+\dbinom{5}{3}x^2y^3+\dbinom{5}{4}x^1y^4+\dbinom{5}{5}x^0y^5 \\ {(x+y)}^5 &= x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5 \end{align*}\] b. Substitute \(n=4\) into the formula. Evaluate the \(k=0\) through \(k=4\) terms. Notice that \(3x\) is in the place that was occupied by \(x\) and that \(–y\) is in the place that was occupied by \(y\). So we substitute them. Simplify. \[\begin{align*} {(3x−y)}^4 &= \dbinom{4}{0}{(3x)}^4{(−y)}^0+\dbinom{4}{1}{(3x)}^3{(−y)}^1+\dbinom{4}{2}{(3x)}^2{(−y)}^2+\dbinom{4}{3}{(3x)}^1{(−y)}^3+\dbinom{4}{4}{(3x)}^0{(−y)}^4 \\ {(3x−y)}^4 &= 81x^4−108x^3y+54x^2y^2−12xy^3+y^4 \end{align*}\] Analysis Notice the alternating signs in part b. This happens because \((−y)\) raised to odd powers is negative, but \((−y)\) raised to even powers is positive. This will occur whenever the binomial contains a subtraction sign. Exercise \(\PageIndex{2}\) Write in expanded form.
\(x^5−5x^4y+10x^3y^2−10x^2y^3+5xy^4−y^5\) Answer b\(8x^3+60x^2y+150xy^2+125y^3\) Using the Binomial Theorem to Find a Single TermExpanding a binomial with a high exponent such as \({(x+2y)}^{16}\) can be a lengthy process. Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term. Note the pattern of coefficients in the expansion of \({(x+y)}^5\). \({(x+y)}^5=x^5+\dbinom{5}{1}x^4y+\dbinom{5}{2}x^3y^2+\dbinom{5}{3}x^2y^3+\dbinom{5}{4}xy^4+y^5\) The second term is \(\dbinom{5}{1}x^4y\). The third term is \(\dbinom{5}{2}x^3y^2\). We can generalize this result. THE \((R+1)\)TH TERM OF A BINOMIAL EXPANSION The \((r+1)\)th term of the binomial expansion of \({(x+y)}^n\) is: \[\dbinom{n}{r}x^{n−r}y^r \label{binomial5}\] How to: Given a binomial, write a specific term without fully expanding.
Example \(\PageIndex{3}\): Writing a Given Term of a Binomial Expansion Find the tenth term of \({(x+2y)}^{16}\) without fully expanding the binomial. Solution Because we are looking for the tenth term, \(r+1=10\), we will use \(r=9\) in our calculations and Equation \ref{binomial5}. \(\dbinom{16}{9}x^{16−9}{(2y)}^9=5,857,280x^7y^9\) Exercise \(\PageIndex{3}\) Find the sixth term of \({(3x−y)}^9\) without fully expanding the binomial. Answer\(−10,206x^4y^5\) Media Access these online resources for additional instruction and practice with binomial expansion.
Key EquationsBinomial Theorem\({(x+y)}^n=\sum_{k=0}^n\dbinom{n}{k}x^{n−k}y^k\)\((r+1)\)th term of a binomial expansion\(\dbinom{n}{r}x^{n−r}y^r\)Key Concepts
Contributors and Attributions
This page titled 13.6: Binomial Theorem is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. What is the nth term of the expansion?The formula to find the nth term in the binomial expansion of (x + y)n is Tr+1 = nCr xn-ryr. Applying this to (2x + 3)9 , T5 = T4+1 = 9C4 (2x)9-434.
What is the formula for expansion?ΔL = αLΔT is the formula for linear thermal expansion, where ΔL is the change in length L, ΔT is the change in temperature, and is the linear expansion coefficient, which varies slightly with temperature.
What is a term in binomial expansion?The top number of the binomial coefficient is n, which is the exponent on your binomial. The bottom number of the binomial coefficient is r - 1, where r is the term number. a is the first term of the binomial and its exponent is n - r + 1, where n is the exponent on the binomial and r is the term number.
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