Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Fundamental theorem of algebra wikipedia , lookup

Universal enveloping algebra wikipedia , lookup

Clifford algebra wikipedia , lookup

Geometric algebra wikipedia , lookup

Exterior algebra wikipedia , lookup

History of algebra wikipedia , lookup

Transcript
```Binomial Expansion and Surds.
Part 1
By
Mr Porter
Assumed Knowledge.
Student should be able to expand a single bracket expression.
Examples: Expand (and simplify)
(
a) 2 3 - 5
(
)
= 2 ´ 3- 5
The meaning of bracket is X (times)
)
Each item in the bracket is to multiplied by the item on the outside , in this case ‘2’.
= 2 ´ 3- 2 ´ 5
The ‘number’ part can be simplified, the √5 behaves like algebra.
= 6-2 5
Note : Number multiply number and surds multiply surds,
but number and surds, behave like algebra.
1
{Assumed Knowledge.}
Examples: Expand
(
b) 2 5 + 2
(
)
= 2 ´ 5+ 2
The meaning of bracket is X (times)
)
Each item in the bracket is to multiplied by the item on the outside , in this case ‘√2’.
= 2 ´5- 2 ´ 2
The ‘surd’ part can be simplified, the (number/surd) behaves like algebra.
=5 2 -2
This is the final answer. You should know √2 X √2 = √4 = 2 or (√2)2 = 2
Note : Number multiply number and surds multiply surds,
but number and surds, behave like algebra.
2
{Assumed Knowledge.}
Examples: Expand
c)
(
2 3- 5 2
(
)
= 2 ´ 3- 5 2
The meaning of bracket is X (times)
)
Each item in the bracket is to multiplied by the item on the outside , in this case ‘√2’.
= 2 ´ 3- 2 ´ 5 2
The ‘surd’ part can be simplified, the √2 x 3 behaves like algebra.
= 3 2 -5´2
This is the final answer. You should know √2 x 5√2 = 5√4 =5 x 2 or (√2)2 = 2
= 3 2 -10
Note : Numbers multiply numbers and surds multiply surds,
but number and surds, behave like algebra.
2
Binomial Product and surds.
This is like combining any two of the examples of single brackets.
Examples: Expand (and simplify)
( 3 + 5 )( 4 + 2 )
Binomial, meaning 2 brackets in this case to be expanded, we use the distributive law.
(There are other methods.)
To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign]
Then , multiply the second bracket by each of the parts of the bracket you split.
(
)
(
= 3´ 4 + 2 + 5 ´ 4 + 2
)
Now, multiply out each of the individual brackets.
= 3´ 4 + 3 ´ 2 + 5 ´ 4 + 5 ´ 2
= 12 + 3 2 + 4 5 + 10
This expansion is finished.
Numbers multiply numbers and surds multiply surds,
but numbers and surds, behave like algebra.
Now, look very HARD at the surd parts. All the
surds are different (just like algebra) and they
cannot be broken down by the square numbers 4, 9,
16, 25, ….
Binomial Product and surds.
Examples: Expand (and simplify)
(
)(
2+ 2 7- 2
(
)
)
Last example was very boring, but necessary.
To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign]
Then , multiply the second bracket by each of the parts of the bracket you split.
(
= 2´ 7- 2 + 2 ´ 7- 2
)
Notice, the un-split bracket is written out twice. Now, multiply out each of
the individual brackets.
= 2´7-2´ 2+ 2 ´7- 2 ´ 2
= 14 - 2 2 + 7 2 - 2
Numbers multiply numbers and surds multiply surds,
but numbers and surds, behave like algebra.
Did you remember that √2 x √2 = √4 = 2 or (√2)2 = 2
Now, look very HARD at the surd parts. All the surds are the same (just like
algebra) and they can be combined.
= 12 + 5 2
This expansion is finished.
Binomial Product and surds are more
interesting, when the surd part is the same!
Binomial Product and surds.
Examples: Expand (and simplify)
To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign]
Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the
parts of the bracket you split.
( 5 - 3 )(1- 4 3 )
(
= 5 ´ 1- 4 3
)
(
- 3 ´ 1- 4 3
)
Notice, the un-split bracket is written out twice. Now, multiply out each of
the individual brackets (take care of negative signs!).
= 5 ´1- 5 ´ 4 3 - 3 ´1- - 3 ´ 4 3
= 5 - 20 3 -1 3 + 4 ´ 3
DOUBLE negative is ‘+’
Did you remember that √3 x √3 = √9 = 3 or (√3)2 = 3
Now, look very HARD at the surd parts. All the surds are the same (just like
algebra) and they can be combined.
= 17 - 21 3
This expansion is finished.
Binomial Product and surds are more
interesting, when the surd part is the same!
Binomial Product and surds.
Examples: Expand (and simplify)
(2
5- 3
)(
(
5+4 3
= 5´ 2 5- 3
)
)
To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’ sign]
Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by each of the
parts of the bracket you split.
(
+4 3´ 2 5 - 3
= 2 ´ 5 - 15 + 8 15 - 4 ´ 3
)
Notice, the un-split bracket is written out twice. Now,
multiply out each of the individual brackets (take care of
negative signs!).
Did you remember that
√5 x √5 = √25 = 5 or (√5)2 = 5
= 10 - 15 + 8 15 +12
√3 x √3 = √9 = 3 or (√3)2 = 3
Now, look very HARD at the surd parts. All the surds are the same
(just like algebra) and they cannot be broken down by the square
numbers 4, 9, 16, 25, ….
= 22 + 7 15
This expansion is finished.
Binomial Product and surds are more
interesting, when the surd part is the same!
Binomial Product and surds.
Examples: Expand (and simplify)
(
) (
2
5-3 =
= 5´
(
)(
5-3
)
5-3
- 3´
To apply the distributive law, split one of the brackets. [Usually, one containing a ‘+’
sign]
Otherwise, the bracket with the simplest surd. Then , multiply the second bracket by
each of the parts of the bracket you split.
)
5-3
(
= 5 - 3 5 - 3 5 + 3´ 3
(….)2 = (…..) (…..), the first step is the remove the power ‘2’.
)
5-3
Notice, the un-split bracket is written out twice. Now, multiply out
each of the individual brackets (take care of negative signs!).
Did you remember that
√5 x √5 = √25 = 5 or (√5)2 = 5
=5 -3 5-3 5 +9
= 14 - 6 5
Now, look very HARD at the surd parts. All the surds are the same
(just like algebra) and they cannot be broken down by the square
numbers 4, 9, 16, 25, ….
This expansion is finished.
Binomial Product and surds are more interesting, when the
surd part is the same!
Binomial Product and surds.
Examples: Expand (and simplify)
(2
) (
)(
2
5+ 3 = 2 5+ 3 2 5+ 3
(
=2 5´ 2 5+ 3
(
= 2 5
)
2
)
To apply the distributive law, split one of the brackets. [Usually, one
containing a ‘+’ sign]
Otherwise, the bracket with the simplest surd. Then , multiply the second
bracket by each of the parts of the bracket you split.
)
+ 3´ 2 5 + 3
+ 2 15 + 2 15 +
= 20 + 2 15 + 2 15 + 3
= 23 + 4 15
(
(….)2 = (…..) (…..), the first step is the remove the power ‘2’.
( 3)
2
)
Notice, the un-split bracket is written out twice. Now,
multiply out each of the individual brackets (take care of
negative signs!).
Did you remember that
2√5 x 2√5 = 4√25 = 20 or (2√5)2 = 4 x 5 = 20
√3 x √3 = √9 = 3
or (√3)2 = 3
Now, look very HARD at the surd parts. All the surds are the same
(just like algebra) and they cannot be broken down by the square
numbers 4, 9, 16, 25, ….
This expansion is finished.
Binomial Product and surds are more interesting, when the
surd part is the same!
```