9.5 The Binomial Theorem
Binomial is a polynomial that has 2 terms
(x + y)0 = 1
(x + y)1 = 1x + 1y
(x + y)2 = 1x2 +2xy + 1y2
(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3
(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4
(x + y)5 = 1x5 + 5x4y + 10 x3y2 + 10 x2y3 + 5xy4 + 1y5
Notice the pattern that is developing in each expansion of the binomial.
1. The sum of the exponents of each term is “n”.
2. In each expression, there are n + 1 terms.
3. In each expansion, x and y have symmetric roles. The powers of x decreases by 1 in successive terms, whereas the powers of y increase by 1.
4. The coefficient increase and decrease in a symmetric pattern.
The coefficients of a binomial expansion are called binomial coefficients.
A display that contains only the coefficients of the terms in the expansions is called Pascal’s Triangle. The first triangle is Pascal’s Triangle, the second is using combinations to find the numbers.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
0C0
1C0 1C1
2C0 2C1 2C2
3C0 3C1 3C2 3C3
4C0 4C1 4C2 4C3 4C4
5C0 5C1 5C2 5C3 5C4 5C5
Therefore in general, The Binomial Theorem: In the expansion of
(x + y)n = nC0 (x)n y0 + nC1 (x)n-1 y1 + nC2 (x)n-2 y2 + ... + nCn-1 (x)1 yn-1 + nCn (x)0 yn
The coefficient of xn-r yr is
The symbol ( ) is often used in place of nCrto denote binomial coefficients.
Example 1: Find the following pairs of binomial coefficients.
a. 7C0 , 7C7 = 1 , 1
b. 7C1 , 7C6= 7 , 7
c. 8C2 , 8C6 = 28 , 28 same answer
What can you conclude?
nCr = nCn - r
This shows the symmetric property of binomial coefficients.
(2x - 3y)3 = 3C0 (2x)3(-3y)0 + 3C1 (2x)2 (-3y)1 + 3C2(2x)1(-3y)2 + 3C3 (2x)0(-3y)3
= (1)(8x3)(1) + (3)(4x2)(-3y) + (3)(2x)(9y2) + (1)(1)(-27y3)
= 8x3 - 36x2y + 54xy2 - 27y3
Remember to put the negative sign in the parenthesis so when you raise it to a power, the answer is appropriately positive or negative!!
Expand (2x - y)5 = (2x)5 + 5(2x)4(-y) + 10(2x)3(-y)2 + 10(2x)2(-y)3 + 5(2x)(-y)4 + (-y)5
= 32x5 - 80x4y + 80 x3y2 - 40 x2y3 + 10xy4- y5
Do you notice a pattern for (x - y)n ?
