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CONTENTS

BINOMIAL THEOREM

FERMAT’S THEOREM

WILSON’S THEOREM WILSON’S THEOREM

DI-PHANTIUM THEOREM

n

x a n

x a

1

x a x a

A binomial is a polynomial with two terms such as x + a. Often we need to raise a binomial to a power. In this section we'll explore a way to do just that without lengthy multiplication.

2 2 22x a x ax a

3 3 2 2 33 3x a x ax a x a

0

1x a

3 3 2 2 33 3x a x ax a x a

4 4 3 2 2 3 44 6 4x a x ax a x a x a

5

x a 5 4 2 3 3 2 4 5__ __ __ __x ax a x a x a x a

We can easily see the pattern on the x's and the a's. But what about the coefficients? We’ll see that in our next slide.

Let's list all of the coefficients on the x's and the a's and look for a pattern.

0

1

1 1

1 2 1

+

+ +

5

x a 5 4 2 3 3 2 4 51 5 10 10 5 1x ax a x a x a x a

1

1 1x a x a

2 2 21 2 1x a x ax a

3 3 2 2 31 3 3 1x a x ax a x a

4 4 3 2 2 3 41 4 6 4 1x a x ax a x a x a

0

1x a 1 3 3 1

1 4 6 4 1

+ + +

1 5 10 10 5 1

+ + + +

1

1 1

1 2 1

1 3 3 11 3 3 1

1 4 6 4 1

1 5 10 10 5 1

This is called Pascal's Triangle and would give us the coefficients for a binomial expansion of any power if we extended it far enough.

! !The Factorial Symbol

0! = 1 1! = 1n! = n(n-1) · . . . · 3 · 2 · 1 n must be an integer greater than or equal to 2

!!

What this says is if you have a positive integer followed by the factorial symbol you multiply the integer by each integer less than it until you get down to 1.

6! = 6 · 5 · 4 · 3 · 2 · 1 = 720

If and are integers with 0 ,

the symbol is defined as

!

j n j n

n

j

n n

!

! !

n n

j j n j

This symbol is read "n taken j at a time"

1

0 1

n n n nn n nx a x ax a

n

The Binomial Theorem

The x's start out to the nth power and decrease by 1 in power

4 8495a x

The x's start out to the nth power and decrease by 1 in power

each term. The a's start out to the 0 power and increase by 1 in power each term. The binomial coefficients are found by computing the combination symbol. Also the sum of the

powers on a and x is n.

The 5th term of (x + a)12

5th term will have a4

(power on a is 1 less than term number)

So we'll have x8

(sum of two powers is 12)

4 812

4a x

1 less

than term number

Here is the expansion of (x + a)12

…and the 5th term matches the term we obtained!

In this expansion, observe the following:

•Powers on a and x add up to power on binomial•Powers on a and x add up to power on binomial

•a's increase in power as x's decrease in power from term to term.

•Powers on a are one less than the term number

•Symmetry of coefficients (i.e. 2nd term and 2nd to last term

have same coefficients, 3rd & 3rd to last etc.) so once you've reached the middle, you can copy by symmetry rather than compute coefficients.

First let's write out the expansion of the general (x + a)6 and then we'll substitute.

6 6 5 2 4 3 3 4 2 5 6__ __ __ __ __x a x ax a x a x a x a x a

these will be the same

6 615 1520

6 6 5 2 4

6 152 2 2 23 3 3y y yx x x x

EXAMPLE: TO EXPAND (2x - 3y)^6

these will be the same

these will be the same

Let's find the coefficient for the second term.

6 6! 6 5!6

1 1!5! 5!

Let's confirm that this is also the coefficient of the 2nd to last term.

6 6! 6 5!6

5 5!1! 5!

Let's find the coefficient for the third term.

6 6! 6 5 4!15

2 2!4! 2 4!

This will also be the coefficient of the 3rd to last term.

Now we'll find the coefficient of the 4th term

6 6! 6 5 4 3!20

3 3!3! 3 2 3!

Now we'll apply this formula to our specific binomial.

3 3 4 2 5 6

6 15

20

2 2 2 2

2 215

3 3 3

3 3 3 36 2

y y y

y y y

x x x

x x y

x

x

6 5 4 2 3 3

2 4 5 6

64 576 2160 4320

4860 2916 729

x x y x y x y

x y xy y

FERMAT’S THEOREM

A Little Bit Of HistoryPIERRE DE FERMAT (1601-1665) WAS A LAWYER BYPROFESSION AND AN AMATEUR MATHEMATICIAN. FERMATRARELY PUBLISHED HIS MATHEMATICAL DISCOVERIES. ITWAS MOSTLY THROUGH HIS CORRESPONDENCE WITHOTHER MATHEMATICIANS THAT HIS WORK IS KNOWN ATOTHER MATHEMATICIANS THAT HIS WORK IS KNOWN ATALL. FERMAT WAS ONE THE INVENTORS OF ANALYTICGEOMETRY AND CAME UP WITH SOME OF THEFUNDAMENTAL IDEAS OF CALCULUS. HE IS PROBABLYMOST FAMOUS FOR A PROBLEM THAT WENT UNSOLVED

UNTIL 1994; THAT THE EQUATION xn + yn = zn HAS

NO NON-TRIVIAL SOLUTION WHEN n>2.

FERMAT’S THEOREM

IF ‘P’ IS A PRIME NUMBER AND ‘N’ IS PRIME TO ‘P’ THEN NP-1-1 IS A MULTIPLE OF ‘P’.

Proof Proof

we know that (α+β)p = αp + βp + M(p)

consider (b+c+d+….)p = βp

then we can prove that

(a+b+c+d+…..)p = ap +bp + cp + dp +…+ M(p)

Proof continued…

Let each of the quantities a,b,c,d,… be equal to

unity, and suppose they are N in number; then

Np = N+M(p)

Np – N = M(p)

N(Np-1 – 1) = M(p).

But ‘N’ is prime to ‘p’, and therefore Np-1 – 1 is a multiple of ‘p’.

DIOPHANTINE THEOREM

ax + by = gcd(a,b)

Suppose ‘a’ and ‘b’ are integers, then The Diophantine equation ax+by=gcd(a,b) has a solution

Example: Find a solution to the diophantine equation 47x+30y=1

Remember, if you divide a by b you get a quotient q and a remainder r.

They satisfy a=bq+r

47=30(1)+17

30=17(1)+1330=17(1)+13

17=13(1)+4

13=4(3)+1

Solve for the remainders

47=30(1)+17 » 17=47(1)+30(-1)

30=17(1)+13 »13=30(1)+17(-1)

17=13(1)+4 » 4=17(1)+13(-1)

13=4(3)+1 » 1=13(1)+4(-3)

47=30(1)+17 » 17=47(1)+30(-1)

30=17(1)+13 »13=30(1)+17(-1)

17=13(1)+4 » 4=17(1)+13(-1)

13=4(3)+1 » 1=13(1)+4(-3)

1 = 13(1)+4(-3) substitute for 4= 13(1)+(17(1)+13(-1))(-3)= 13(1)+17(-3)+13(3)= 13(4)+17(-3) substitute for 13= (30(1)+17(-1))(4)+17(-3)= 30(4)+17(-4)+17(-3)= 30(4)+17(-4)+17(-3)= 30(4)+17(-7) substitute for 17= 30(4)+(47(1)+30(-1)0(-7)= 30(4)+47(-7)+30(7)= 30(11)+47(-7) Done!!

So the equation 47x +30y = 1 has a solution x=-7, y=11.

WILSON’S THEOREMIf ‘n’ is a prime number, (n-1)!+1 is divisible by ‘n’.

EXAMPLE:

lets take n=5

then (n-1)!+1 = 4!+1

=24+1

=25 (which is divisible by 5)

USE OF THIS THEOREM

We can test the divisibility of numbers by this process.

For example: Given to find if 721 is divisible by 7 or not.by 7 or not.

we can write 721 as » 720+1

» 6! + 1

» (7-1)!+1

(which is divisible by 7)