5.2 Binomial Theorem A14.notebook
May 05, 2014
Binomial Theorem
t0,0
t1,0 t1,1 t2,0 t2,1 t2,2
t3,0 t3,1 t3,2 t3,3
t4,0 t4,1 t4,2 t4,3 t4,4
1
0C0
1 1
1C0 1C1 1 2 1
2C0 2C1 2C2 3C0 3C1 3C2 3C3
1 3 3 1
1 4 6 4 1 4C0 4C1 4C2 4C3 4C4
* Each term in Pascal's Triangle corresponds to a value of nCr
Binomial Expansion
Recall: Binomial (2 terms)
Ex. (x+y), (3x+8), (2x2­1)
Expand a) (a+b)1
= a + b
b) (a+b)2
=(a+b)(a+b)
= a2+ab+ab+b2
= a2+2ab+b2
c) (a+b)3
=(a+b)(a+b)(a+b)
=(a2+2ab+b2)(a+b)
=a3+3a2b+3ab2+b2
* The coefficients of each term in the expansion of (a+b) n corresponds to the terms in row ' n' of Pascal's Triange
5.2 Binomial Theorem A14.notebook
May 05, 2014
Binomial Theorem
(a+b)n = nC0an + nC1an­1 b1 + nC2an­2 b2 + nC3an­3 b3 + ...
+ nCran­rbr + nCnbn
* Exponents on the 1st variable start at 'n' and decrease by 1 each term
* Exponents on the 2nd variable start at 0 and increase by 1 each term
* Coefficient starts at nCr, and the number you are 'choosing' (nC_) starts at 0 and increases by one for each term
Ex. Expand, using binomial theorem
a) (a+b)5
= 5C0a5b0 + 5C1a4b1 + 5C2a3b2 + 5C3a2b3 + 5C4a1b4 + 5C5a0b5
= 1a5 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1b5
b) (2x­1)4
= 4C0(2x)4+4C1(2x)3(­1)1+4C2(2x)2(­1)2+4C3(2x)1(­1)3+4C4(­1)4
= 1(16x4) + 4(8x3)(­1) + 6(4x2)(1) + 4(2x)(­1) + 1(1)
= 16x4 ­ 32x3 + 24x2 ­ 8x + 1
c) (3x­2y)3
= 3C0(3x)3 + 3C1(3x)2(­2y)1 + 3C2(3x)1(­2y)2 + 3C3(­2y)3
= 1(27x3) + 3(9x2)(­2y) + 3(3x)(4y2) +1(­8y3)
= 27x3 ­ 54x2y + 36xy2 ­ 8y3
5.2 Binomial Theorem A14.notebook
May 05, 2014
For (x + y)n, the general term is:
This formula can be used to determine the (r + 1)st term in a
binomial expansion.
Remember pascal's
triangle starts at row 0.
Example 1:
In the expansion of (2x+5)9 determine the 4 th term in the
polynomial.
t r+1 = t4
Example 3:
In the expansion of (x + 3)12 determine the coefficient
of the x5 term.
∴
r=3
5.2 Binomial Theorem A14.notebook
May 05, 2014
Example 4:
Find the term containing
in
since this binomial has "x" in both terms the concept used in ex 3 will not work.
Example 5 : Find the constant term (contains
Practise:
p. 293 #2, 3,
9ace, 10, 11ace, 12
) in