TMUA: Binomial expansion
The binomial expansion for positive integer powers.
- Section 1 Part 1 · MM2 Sequences and series · spec MM2.4
- Papers 1 and 2
- 8 practice questions
- No calculator
What the specification covers
- (1 + x)^n and (a + f(x))^n for positive integer n
- n! and nCr; finding a specific coefficient
- Terms independent of x; products of two expansions
Key ideas
- \((1+x)^n=\sum_{r=0}^{n}\binom nr x^r\), with \(\binom nr=\dfrac{n!}{r!(n-r)!}\).
- The general term of \((a+b)^n\) is \(\binom nr a^{n-r}b^r\): use it to find one coefficient without expanding everything.
- For a product such as \((1+2x)^6(1-x)^2\), the \(x^k\) coefficient is a sum of products of coefficients whose powers add to \(k\).
- A term independent of \(x\) has total power of \(x\) equal to 0.
Common mistakes
- Include the sign and the power of the coefficient: in \((1-2x)^5\) the \(x^3\) term is \(\binom53(-2)^3x^3\).
- \(\binom nr=\binom n{n-r}\): don't double count.
Exam tip
Write the general term with the power of \(x\) as a function of \(r\), then solve for \(r\).
Worked example
Worked example
What is the coefficient of \(x^4\) in the expansion of \((1+2x)^6(1-x)^2\)?
- \(-20\)
- \(-80\)
- \(20\)
- \(240\)
- \(100\)
Answer: A: \(-20\)
In \((1+2x)^6\) the coefficients of \(x^2,x^3,x^4\) are \(\binom62 2^2=60\), \(\binom63 2^3=160\), \(\binom64 2^4=240\). With \((1-x)^2=1-2x+x^2\), the \(x^4\) coefficient is \(240\cdot1+160\cdot(-2)+60\cdot1=-20\).
Practice questions
Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.
Question 1
What is the term independent of \(x\) in the expansion of \(\left(x^2-\dfrac{2}{x}\right)^6\)?
- \(-240\)
- \(-160\)
- \(60\)
- \(240\)
- \(160\)
Show the answer and solution
Answer: D: \(240\)
The general term is \(\binom6r(x^2)^{6-r}\left(-\tfrac2x\right)^r=\binom6r(-2)^rx^{12-3r}\). Independent of \(x\) when \(r=4\): \(\binom64(-2)^4=15\times16=240\).
Question 2
What is the coefficient of \(x^{2}\) in the expansion of \((1+2x)^{6}(1-x)^2\)?
- \(37\)
- \(-37\)
- \(48\)
- \(60\)
- \(85\)
Show the answer and solution
Answer: A: \(37\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{6}{j}(2)^j\) is the coefficient of \(x^j\) in \((1+2x)^{6}\), the required coefficient is \(c_{2}-2c_{1}+c_{0}=60-2(12)+1=37\).
Question 3
What is the coefficient of \(x^{4}\) in the expansion of \((1-2x)^{6}(1-x)^2\)?
- \(620\)
- \(400\)
- \(-620\)
- \(240\)
- \(-20\)
Show the answer and solution
Answer: A: \(620\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{6}{j}(-2)^j\) is the coefficient of \(x^j\) in \((1-2x)^{6}\), the required coefficient is \(c_{4}-2c_{3}+c_{2}=240-2(-160)+60=620\).
Question 4
What is the coefficient of \(x^{4}\) in the expansion of \((1+2x)^{5}(1-x)^2\)?
- \(0\)
- \(80\)
- \(40\)
- \(280\)
- \(-40\)
Show the answer and solution
Answer: E: \(-40\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{5}{j}(2)^j\) is the coefficient of \(x^j\) in \((1+2x)^{5}\), the required coefficient is \(c_{4}-2c_{3}+c_{2}=80-2(80)+40=-40\).
Question 5
What is the coefficient of \(x^{4}\) in the expansion of \((1+2x)^{4}(1-x)^2\)?
- \(24\)
- \(-24\)
- \(-16\)
- \(16\)
- \(104\)
Show the answer and solution
Answer: B: \(-24\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{4}{j}(2)^j\) is the coefficient of \(x^j\) in \((1+2x)^{4}\), the required coefficient is \(c_{4}-2c_{3}+c_{2}=16-2(32)+24=-24\).
Question 6
What is the coefficient of \(x^{3}\) in the expansion of \((1+3x)^{5}(1-x)^2\)?
- \(465\)
- \(-105\)
- \(270\)
- \(180\)
- \(105\)
Show the answer and solution
Answer: E: \(105\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{5}{j}(3)^j\) is the coefficient of \(x^j\) in \((1+3x)^{5}\), the required coefficient is \(c_{3}-2c_{2}+c_{1}=270-2(90)+15=105\).
Question 7
What is the coefficient of \(x^{2}\) in the expansion of \((1+3x)^{6}(1-x)^2\)?
- \(117\)
- \(135\)
- \(172\)
- \(-100\)
- \(100\)
Show the answer and solution
Answer: E: \(100\)
\((1-x)^2=1-2x+x^2\). If \(c_j=\binom{6}{j}(3)^j\) is the coefficient of \(x^j\) in \((1+3x)^{6}\), the required coefficient is \(c_{2}-2c_{1}+c_{0}=135-2(18)+1=100\).
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