The-Binomial-Expansion-二次項展開式

1、The Binomial ExpansionIFY Maths 1Learning OutcomesExpand for small positive integer nUse Pascals triangle to find the binomial coefficientsExpand for small positive integer nPowers of a + bIn this presentation we will develop a formula to enable us to find the terms of the expansion ofwhere n is any

2、 positive integer.We call the expansion binomial as the original expression has 2 parts.Powers of a + bWe know thatso the coefficients of the terms are 1, 2 and 1We can write this as12112Powers of a + b121121Powers of a + b121Powers of a + b1213311Powers of a + bso the coefficients of the expansion

3、of are 1, 3, 3 and 11211213311Powers of a + b1331133164141Powers of a + b1331336414111This coefficient . . . . . . is found by adding 3 and 1; the coefficients that are in 314Powers of a + b1333614111This coefficient . . . . . . is found by adding 3 and 1; the coefficients that are in Powers of a +

4、bSo, we now haveCoefficientsExpression121133114641So, we now haveCoefficientsExpression121133114641Each number in a row can be found by adding the 2 coefficients above it.Powers of a + bPowers of a + bSo, we now haveCoefficientsExpression121133114641The 1st and last numbers are always 1.Each number

5、in a row can be found by adding the 2 coefficients above it.Powers of a + bSo, we now haveCoefficientsExpression12113311114641To make a triangle of coefficients, we can fill in the obvious ones at the top.1Powers of a + bThe triangle of binomial coefficients is called Pascals triangle, after the Fre

6、nch mathematician. . . but its easy to know which row we want as, for example,starts with 1 3 . . . will start 1 10 . . .Notice that the 4th row gives the coefficients of ExerciseFind the coefficients in the expansion of We usually want to know the complete expansion not just the coefficients.Powers

7、 of a + be.g. Find the expansion of Pascals triangle gives the coefficientsSolution: 15101105The full expansion isTip: The powers in each term sum to 5151010511e.g. 2 Write out the expansion of in ascending powers of x. Powers of a + bSolution:The coefficients area a a a a b b b b bTo get we need to

8、 replacea by 1 ( Ascending powers just means that the 1st term must have the lowest power of x and then the powers must increase. )14614We know that1 1(1)(1)(1) b b b b be.g. 2 Write out the expansion of in ascending powers of x. 14614We know thatPowers of a + bSolution:The coefficients areTo get we

9、 need to replacea by 1Be careful! The minus sign . . .is squared as well as the x.The brackets are vital, otherwise the signs will be wrong!e.g. 2 Write out the expansion of in ascending powers of x. 14614We know thatPowers of a + bSolution:The coefficients areTo get we need to replacea by 1 andb by

10、 (- x)1(1) 1(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifying givesTo get we need to replacea by 1 andb by (- x)Since we know that any power of 1 equals 1, we could have written 1 here . . . e.g. 2 Write out the expansion of in ascending powers of x. 14614We know thatPowers of a + bSolution:The coefficients are

11、1 1(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifying givesTo get we need to replacea by 1 andb by (- x)e.g. 2 Write out the expansion of in ascending powers of x. 14614We know thatPowers of a + bSolution:The coefficients are1 1(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifying givesSince we know that any power of 1 eq

12、uals 1, we could have written 1 here . . . To get we need to replacea by 1 andb by (- x)e.g. 2 Write out the expansion of in ascending powers of x. 14614We know thatPowers of a + bSolution:The coefficients are1 1(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifying gives. . . and missed these 1s out.e.g. 2 Write

13、 out the expansion of in ascending powers of x. 14614We could go straight toPowers of a + bSolution:The coefficients are1 1(-x)(-x)(-x)(-x)(-x)Simplifying givesExercise1. Find the expansion of in ascending powers of x.Powers of a + bIf we want the first few terms of the expansion of, for example, ,

14、Pascals triangle is not helpful.We will now develop a method of getting the coefficients without needing the triangle.Each coefficient can be found by multiplying the previous one by a fraction. The fractions form an easy sequence to spot.Powers of a + bLets considerWe know from Pascals triangle tha

15、t the coefficients are1615115620There is a pattern here:So if we want the 4th coefficient without finding the others, we would need( 3 fractions )Powers of a + bThe 9th coefficient of isFor we get1201901140etc.Even using a calculator, this is tedious to simplify. However, there is a shorthand notati

16、on that is available as a function on the calculator.Powers of a + bWe write 20 !is called 20 factorial.( 20 followed by an exclamation mark )We can writeThe 9th term of is Powers of a + bcan also be written asorThis notation. . . . . . gives the number of ways that 8 items can be chosen from 20. is

17、 read as “20 C 8” or “20 choose 8” and can be evaluated on our calculators.The 9th term of is then In the expansion, we are choosing the letter b 8 times from the 20 sets of brackets that make up . ( a is chosen 12 times ).Powers of a + bThe binomial expansion of isWe know from Pascals triangle that

18、 the 1st two coefficients are 1 and 20, but, to complete the pattern, we can write these using the C notation:and Since we must define 0! as equal to 1. Powers of a + bTip: When finding binomial expansions, it can be useful to notice the following:So, is equal to Any term of can then be written as where r is any integer from 0 to 20.The expansion of is Any term of can be written in the form where r is any integer from 0