The Binomial Expansion 二次项展开式

1、The Binomial Expansion,Learning Outcomes,Expand for small positive integer n Use Pascals triangle to find the binomial coefficients Expand for small positive integer n,Powers of a + b,We call the expansion binomial as the original expression has 2 parts.,Powers of a + b,We know that,so the coefficie

2、nts of the terms are 1, 2 and 1,We can write this as,1,2,Powers of a + b,1,2,1,Powers of a + b,Powers of a + b,Powers of a + b,so the coefficients of the expansion of are 1, 3, 3 and 1,1,2,1,1,2,1,3,3,1,1,Powers of a + b,Powers of a + b,1,3,3,1,3,3,6,4,1,4,1,1,1,This coefficient . . .,. . . is found

3、 by adding 3 and 1; the coefficients that are in,3,1,4,Powers of a + b,1,3,3,3,6,1,4,1,1,1,This coefficient . . .,. . . is found by adding 3 and 1; the coefficients that are in,Powers of a + b,So, we now have,Coefficients,Expression,So, we now have,Coefficients,Expression,Each number in a row can be

4、 found by adding the 2 coefficients above it.,Powers of a + b,Powers of a + b,So, we now have,Coefficients,Expression,The 1st and last numbers are always 1.,Each number in a row can be found by adding the 2 coefficients above it.,Powers of a + b,So, we now have,Coefficients,Expression,To make a tria

5、ngle of coefficients, we can fill in the obvious ones at the top.,1,Powers of a + b,The triangle of binomial coefficients is called Pascals triangle, after the French mathematician.,. . . but its easy to know which row we want as, for example,starts with 1 3 . . .,will start 1 10 . . .,Notice that t

6、he 4th row gives the coefficients of,Exercise,Find the coefficients in the expansion of,We usually want to know the complete expansion not just the coefficients.,Powers of a + b,e.g. Find the expansion of,The full expansion is,1,e.g. 2 Write out the expansion of in ascending powers of x.,Powers of a

7、 + b,To get we need to replacea by 1,( Ascending powers just means that the 1st term must have the lowest power of x and then the powers must increase. ),We know that,1,1,(1),(1),(1),b,b,b,b,b,e.g. 2 Write out the expansion of in ascending powers of x.,We know that,Powers of a + b,Solution:,The coef

8、ficients are,To get we need to replacea by 1,is squared as well as the x.,e.g. 2 Write out the expansion of in ascending powers of x.,We know that,Powers of a + b,Solution:,The coefficients are,To get we need to replacea by 1 and b by (- x),1,(1),1,(1),(1),(-x),(-x),(-x),(-x),(-x),Simplifying gives,

9、To get we need to replacea by 1 and b by (- x),e.g. 2 Write out the expansion of in ascending powers of x.,We know that,Powers of a + b,Solution:,The coefficients are,Simplifying gives,To get we need to replacea by 1 and b by (- x),e.g. 2 Write out the expansion of in ascending powers of x.,We know

10、that,Powers of a + b,Solution:,The coefficients are,Simplifying gives,To get we need to replacea by 1 and b by (- x),e.g. 2 Write out the expansion of in ascending powers of x.,We know that,Powers of a + b,Solution:,The coefficients are,Simplifying gives,e.g. 2 Write out the expansion of in ascendin

11、g powers of x.,We could go straight to,Powers of a + b,Solution:,The coefficients are,Simplifying gives,Exercise,1. Find the expansion of in ascending powers of x.,Powers of a + b,If we want the first few terms of the expansion of, for example, , Pascals triangle is not helpful.,We will now develop

12、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 + b,Lets consider,We know from Pascals triangle that the coefficients are,There is a pattern here:,

13、So if we want the 4th coefficient without finding the others, we would need,( 3 fractions ),Powers of a + b,The 9th coefficient of is,For we get,1,20,190,1140,etc.,Even using a calculator, this is tedious to simplify. However, there is a shorthand notation that is available as a function on the calc

14、ulator.,Powers of a + b,We write 20 !,is called 20 factorial.,( 20 followed by an exclamation mark ),We can write,The 9th term of is,Powers of a + b,can also be written as,or,This notation. . .,. . . gives the number of ways that 8 items can be chosen from 20.,is read as “20 C 8” or “20 choose 8” an

15、d 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 + b,The binomial expansion of is,We know from Pascals triangle that the 1st two coefficients are 1 a

16、nd 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 + b,So, is equal to,The expansion of is,Any term of can be written in the form,where r is any integer from 0 to n.,Generalizations,The binomial expansion of in ascending powers of x is given by,e.g.3 Find the first 4 terms in the expansion of in ascending powers of x.,Powers of a + b,Solution:,e.g.4 Find the 5th term of the expansion of in ascending powers of x.,Solution: The 5th term contains,Powers of a + b,It is,The binomial exp