- Binomial Expansion

For any positive integer \(n\):

\[\displaystyle(a+b)^n=\sum_{r=0}^n \:^nC_r a^{n-r} b^r\]
\[(a+b)^n=\binom{n}{0}a^n+\binom{n}{1}a^{n-1}b^1+\binom{n}{2}a^{n-2}b^2+\binom{n}{3}a^{n-3}b^3+⋯+\binom{n}{n-1}a^1 b^{n-1}+\binom{n}{n}a^0 b^n\]

LESSON 1

Determine the expansion \((3+2x)^5\).

SOLUTION
\[(3+2x)^5=\binom{5}{0}3^5+\binom{5}{1}3^4(2x)^1+\binom{5}{2}3^3(2x)^2+\binom{5}{3}3^2(2x)^3+\binom{5}{4}3^1(2x)^4+\binom{5}{5}(2x)^5\]
\[(3+2x)^5=243+5[81(2x)]+10[27(4x^2 )]+10[9(8x^3)]+5[3(16x^4)]+1(32x^5)\]
\[(3+2x)^5=243+810x+1080x^2+720x^3+240x^4+32x^5\]


LESSON 2

Find the 6th term of the expansion \((2-3x)^{10}\).

SOLUTION
The 6th term of the expansion \((2-3x)^{10}\) occurs when \(r=5\) since the summation index begins with \(r=0\).
\[\binom{10}{5}2^5(-3x)^5\]
\[⟹252(32)(-243x^5)\]
\[⟹-1959552x^5\]

TRY THIS

1. Find the binomial expansion of \((2-x)^3\).

\([8-12x+6x^2-x^3]\)
2. Find the binomial expansion of \(\displaystyle(x+{5\over x})^3\), simplifying the terms.
\([x^3+{125\over x^3}+15x+{75\over x}]\)
3. Expand \(\displaystyle(1+{x\over2})^4\), simplifying the coefficients.
\([{x^4\over16}+{x^3\over2}+{3x^2\over2}+2x+1]\)
4. Find the binomial expansion of \(\displaystyle(x^2+{1\over x})^3\), simplifying your answer.
\([x^6+3x^3+{1\over x^3}+3]\)
5. Calculate the coefficient of \(x^4\) in the expansion of \((x+5)^6\).
[375]
6. Find the coefficient of \(x^3\) in the expansion of \((2+3x)^5\).
[720]
7. Find the coefficient of \(x^3\) in the expansion of \((1-2x)^6\).
\([-160]\)
8. Find the coefficient of \(x^3\) in the expansion of \((3-2x)^5\).
\([-720]\)
9. Find the coefficient of \(x^3\) in the binomial expansion \(\displaystyle(1-{x\over2})^8\).
\([-7]\)

LESSON 3

Find the coefficient of \(x^6\) in the expansion of \((1+3x)^2(2-x)^6\).
SOLUTION
Expand \((1+3x)^2\)
\[(1+3x)^2=1+6x+9x^2\]
Determine the last three terms of the expansion of \((2-x)^6\) since we only need the terms which will result in a \(x^6\) term after multiplication.
\((2-x)^6=\binom{6}{4}2^2 (-x)^4+\binom{6}{5}2(-x)^5+\binom{6}{6}(-x)^6\)
\(=15(4) x^4+6(2)(-x^5 )+1(x^6)\)
\(=60x^4-12x^5+x^6\)
We isolate the multiplications which would create an \(x^6\) term.
\((1+6x+9x^2 )(…+60x^4-12x^5+x^6 )=1(x^6 )+6x(-12x^5 )+9x^2 (60x^4)\)
\(=x^6-72x^6+540x^6\)
\(=469x^6\)

TRY THIS

1. (i) Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \((1+2x)^7\).

(ii) Hence find the coefficient of \(x^2\) in the expansion of \(\displaystyle(2-5x)(1+2x)^7\).
(i) \(1+14x+84x^2\) (ii) 98
2. (i) Find an simplify the first four terms in the binomial expansion of \(\displaystyle(1+{x\over2})^{10}\) in ascending powers of \(x\).
(ii) Hence find the coefficient of \(x^3\) in the expansion of \(\displaystyle(3+4x+2x^2)(1+{x\over2})^{10}\).
(i) \(1+5x+{45\over4}x^2+15x^3\) (ii) 100
3. (i) In the binomial expansion of \(\displaystyle(2x-{1\over 2x})^5\), the first three terms are \(32x^5-40x^3+20x\). Find the remaining three terms of the expression.
(ii) Hence find the coefficient of \(x\) in the expansion of \(\displaystyle(1+4x^2)(2x-{1\over2x})^5\).
(i) \({-5\over x}+{5\over 8x^3}-{1\over 32x^5}\) (ii) 0
4. (i) Find the coefficient of \(x\) in the expansion of \((2x-{1\over x})^5\).
(ii) Hence find the coefficient of \(x\) in the expansion of \(\displaystyle(1+3x^2)(2x-{1\over x})^5\).
(i) 80 (ii) \(-40\)
5. (i) Find the first three terms of the expansion of \(\displaystyle(2x-{3\over x})^5\) is descending powers of \(x\).
(ii) Hence find the coefficient of \(x\) in the expansion of \(\displaystyle(1+{2\over x^2})(2x-{3\over x})^5\).
(i) \(32x^5-240x^3+720x\) (ii) 240

LEsson 4

Find the term independent of \(x\) in the expansion \(\displaystyle(3x^3+{1\over 2x})^8\).

SOLUTION
The term independent of \(x\) will occur when the power of the \(x\) term in the numerator and the denominator are the same. This can be determined by inspection.
\[\binom{8}{6}(3x^3 )^2 ({1\over 2x})^6\]
\[⟹28(9x^6)({1\over 64x^6})\]
\[⟹{63\over16}\]

TRY THIS

1. The binomial expansion of \(\displaystyle(2x+{5\over x})^6\) has a term which is a constant. Find this term.

20 000
2. Find the term independent of \(x\) in the expansion of \(\displaystyle(2x+{1\over 4x^2})^6\).
15
3. The coefficient of \(x^2\) in the expansion of \(\displaystyle(4+ax)(1+{x\over2})^6\) is 3. Find the value of the constant \(a\).
\(-4\)
4. The term independent of \(x\) in the expansion of \(\displaystyle(2x+{k\over x})^6\), where \(k\) is a constant, is 540.
(i) Find the value of \(k\).
(ii) For this value of \(k\), find the coefficient of \(x^2\) in the expansion.
(i) \(k={3\over2}\) (ii) 540
5. Find the coefficient of \(x\) in the expansion of \(\displaystyle({2\over x}-3x)^5\).
\(-1080\)
6. Find the coefficient of \(\displaystyle{1\over x^2}\) in the expansion of \(\displaystyle(3x+{2\over 3x^2})^7\).
840
7. Find the coefficient of \(\displaystyle {1\over x^3}\) in the expansion of \(\displaystyle (x-{2\over x})^7\).
\(-672\)