Locate and evaluate a term in a binomial expansion

Finding a target term

Use the general term to connect the term position, the index and the factor powers. Solve for the index, check its range, then evaluate the exact coefficient

1. Start the general-term index at zero Order the expansion by decreasing powers of the first displayed factor, starting at r=0. In that order, the term in position k uses r=k−1. \(\displaystyle T_4\Rightarrow r=3\)
2. Write each exponent in terms of r Write the total exponent of the requested variable in terms of r, combining contributions from both factors before matching the requested power \(\displaystyle \left(x^2+y\right)^5:\quad e_x\left(r\right)=10-2r\)
3. Check the allowed index The index must be a whole number from 0 to n inclusive \(\displaystyle 0\le r\le n,\quad r\in\mathbb{Z}\)
4. Evaluate the exact coefficient Substitute the index into the binomial coefficient and both factor powers \(\displaystyle \binom{5}{3}\times2^{2}\times3^{3}=1080\)
5. Constant and middle terms For a constant term set the total variable exponent to zero. There are n+1 terms: even n gives one middle term at r=n/2; odd n gives two at r=(n−1)/2 and r=(n+1)/2. A target with no integer index in 0..n has no such term

Worked example

Find the term containing x squared in the expansion below

1. \(\displaystyle T_{r+1}=\binom{4}{r}\left(3x\right)^{4-r}\left(2y\right)^r\)Write the general term for the binomial expansion
2. \(\displaystyle 4-r=2\quad\Longrightarrow\quad r=2\)Set the power of x equal to 2 and solve for r.
3. \(\displaystyle \binom{4}{2}\times3^{2}\times2^{2}=6\times9\times4=216\)Substitute r = 2 to calculate the exact coefficient
4. \(\displaystyle T_3=216x^{2}y^{2}\)Since r = 2, the term is in position r + 1 = 3.

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