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  1. Problems
  2. /Find the coefficient of x⁷ in the expansion of (x² + 2/x)⁸.
MediumMathematicsHigh School (11–12)Algebra › Binomial TheoremNumericalA-LevelBoard ExamIB

Find the coefficient of x⁷ in the expansion of (x² + 2/x)⁸.

StemPy LabContributed by StemPy Lab

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

Find the coefficient of x7x^7x7 in the expansion of (x2+2x)8\left(x^2 + \dfrac{2}{x}\right)^8(x2+x2​)8.

Check your answer

Work it out on paper, then type your number. The answer itself stays hidden.

Need a reminder? The key ideaShowHide

The general term is Tr+1=nCr an−rbrT_{r+1} = {}^nC_r\,a^{n-r}b^rTr+1​=nCr​an−rbr. To find the term with a given power of xxx, write that power in terms of rrr and set it equal to the power you want. Solve for rrr, and check that rrr is a whole number from 000 to nnn.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

Step 1. The general term. Tr+1=8Cr x2(8−r) 2r x−r=8Cr 2r x16−3rT_{r+1} = {}^8C_r\,x^{2(8-r)}\,2^r\,x^{-r} = {}^8C_r\,2^r\,x^{16-3r}Tr+1​=8Cr​x2(8−r)2rx−r=8Cr​2rx16−3r.

Step 2. Match the power. 16−3r=716 - 3r = 716−3r=7 gives r=3r = 3r=3, a whole number between 000 and 888.

Step 3. The coefficient. 8C3×23=56×8=448{}^8C_3 \times 2^3 = 56 \times 8 = 4488C3​×23=56×8=448.

Answer: r=3r = 3r=3 and the coefficient is 448448448.

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