## - 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 $$\displaystyle (x-{2\over x})^7$$.
$$-672$$