Презентация, доклад по математике Формула Бинома Ньютона, 10 класс

Содержание

Powers of a + bWe call the expansion binomial as the original expression has 2 parts.

Слайд 1Made by ALEX
P1 Chapter 8.2
«The Binomial Expansion
And Binomial Theorem»

Made by ALEXP1 Chapter 8.2«The Binomial ExpansionAnd Binomial Theorem»

Слайд 2Powers of a + b
We call the expansion binomial as the

original expression has 2 parts.
Powers of a + bWe call the expansion binomial as the original expression has 2 parts.

Слайд 3Powers of a + b
so the coefficients of the terms are

1, 2 and 1

We can write this as

Powers of a + bso the coefficients of the terms are 1, 2 and 1We can write

Слайд 4
Powers of a + b


Powers of a + b

Слайд 5Powers of a + b

Powers of a + b

Слайд 6Powers of a + b
1
2
1
1
2
1
3
3
1
1

Powers of a + b1211213311

Слайд 7Powers of a + b

Powers of a + b

Слайд 8Powers of a + b
1
3
3
1
3
3
6
4
1
4
1
1
1
This coefficient . . .

Powers of a + b1331336414111This coefficient . . .

Слайд 93
1
4
Powers of a + b
1
3
3
3
6
1
4
1
1
1
This coefficient . . .

314Powers of a + b1333614111This coefficient . . .

Слайд 10Powers of a + b
So, we now have
Coefficients
Expression

Powers of a + bSo, we now haveCoefficientsExpression

Слайд 11So, we now have
Coefficients
Expression
Each number in a row can be found

by adding the 2 coefficients above it.

Powers of a + b

So, we now haveCoefficientsExpressionEach number in a row can be found by adding the 2 coefficients above

Слайд 12Powers 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 + bSo, we now haveCoefficientsExpressionThe 1st and last numbers are always 1.Each number in

Слайд 13Powers of a + b
So, we now have
Coefficients
Expression
To make a triangle

of coefficients, we can fill in the obvious ones at the top.

1

Powers of a + bSo, we now haveCoefficientsExpressionTo make a triangle of coefficients, we can fill in

Слайд 14Powers of a + b
The triangle of binomial coefficients is called

Pascal’s triangle, after the French mathematician.

. . . but it’s easy to know which row we want as, for example,

Powers of a + bThe triangle of binomial coefficients is called Pascal’s triangle, after the French mathematician..

Слайд 15Exercise
Solution: We need 7 rows

ExerciseSolution: We need 7 rows

Слайд 16We usually want to know the complete expansion not just the

coefficients.

Powers of a + b

The full expansion is

1

We usually want to know the complete expansion not just the coefficients.Powers of a + bThe full

Слайд 17Powers of a + b
( Ascending powers just means that the

1st term must have the lowest power of x and then the powers must increase. )

We know that

Powers of a + b( Ascending powers just means that the 1st term must have the lowest

Слайд 181
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 coefficients are

To get we need to replace a by 1

1 1(1)(1)(1) b b b b be.g. 2 Write out the expansion of

Слайд 19is 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 replace a by 1 and
b by (- x)

1

(1)

1

(1)

(1)

(-x)

(-x)

(-x)

(-x)

(-x)

Simplifying gives

is squared as well as the x.e.g. 2 Write out the expansion of

Слайд 20To get we need to replace a

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 replace a by 1 and b by (- x)e.g.

Слайд 21To get we need to replace a

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 replace a by 1 and b by (- x)e.g.

Слайд 22To get we need to replace a

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 replace a by 1 and b by (- x)e.g.

Слайд 23e.g. 2 Write out the expansion of

in ascending powers of x.

We could go straight to

Powers of a + b

Solution:

The coefficients are

e.g. 2 Write out the expansion of in ascending powers of x.

Слайд 24Exercise
1. Find the expansion of

in ascending powers of x.
Exercise1. Find the expansion of in ascending powers of x.

Слайд 25Powers of a + b
If we want the first few terms

of the expansion of, for example, (a+b)20, Pascal’s triangle is not helpful.

We will now develop a method of getting the coefficients without needing the triangle.

We know from Pascal’s triangle that the coefficients are

Powers of a + bIf we want the first few terms of the expansion of, for example,

Слайд 26Each coefficient can be found by multiplying the previous one by

a fraction. The fractions form an easy sequence to spot.

Powers of a + b







There is a pattern here:

So if we want the 4th coefficient without finding the others, we would need

( 3 fractions )

Each coefficient can be found by multiplying the previous one by a fraction. The fractions form an

Слайд 27Powers of a + b
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 calculator.
Powers of a + b1201901140etc.Even using a calculator, this is tedious to simplify. However, there is a

Слайд 28Powers of a + b
We write 20 !
is called 20 factorial.
(

20 followed by an exclamation mark )

We can write

Powers of a + bWe write 20 !is called 20 factorial.( 20 followed by an exclamation mark

Слайд 29Powers of a + b
can also be written as
or
This notation. .

.

. . . gives the number of ways that 8 items can be chosen from 20.

Powers of a + bcan also be written asorThis notation. . . . . . gives the

Слайд 30Powers of a + b
We know from Pascal’s triangle that the

1st two coefficients are 1 and 20, but, to complete the pattern, we can write these using the C notation:
Powers of a + bWe know from Pascal’s triangle that the 1st two coefficients are 1 and

Слайд 31Powers of a + b

Powers of a + b

Слайд 32where r is any integer from 0 to n.
Generalizations 归纳概括

where r is any integer from 0 to n.Generalizations 归纳概括

Слайд 33Powers of a + b

Powers of a + b

Слайд 34Powers of a + b
It is

Powers of a + bIt is

Слайд 35SUMMARY

SUMMARY

Слайд 36Exercise
Solution:
Solution:

ExerciseSolution:Solution:
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