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| Algebraic
Expressions |
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The cube of binomial |
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The binomial expansion
algorithm |
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The difference of two
squares, multiplying |
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The difference of two
squares, factoring |
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The difference of two
cubes |
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The sum of two cubes |
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| Cube
of a binomial |
| To the
cube of the first term add, three times the product of the
square of the first term and the last term, three times the
product of the first term and the square of the last term, and
the cube of the last term. |
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Examples:
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a) (a - b)3
= (a
- b)2
·
(a - b)
= (a2
- 2ab
+
b2)
·
(a - b)
= |
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= a3 - 2a2b
+
ab2
- a2b
+
2ab2
- b3
= a3 - 3a2b
+
3ab2
- b3, |
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b) (x - 2)3
= x3 +
3 ·
x2
·
(-2)
+ 3
·
x ·
(-2)2
+
(-2)3
= x3 - 6x2
+
12x
- 8, |
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c) (2x
+ y)3
= (2x)3 +
3 ·
(2x)2
· y
+ 3
·
(2x)
· y2 + y3
= 8x3 +
12x2y
+
6xy2
+ y3. |
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| The binomial expansion
algorithm - the binomial theorem |
| The
binomial expansion of any positive integral power of a binomial,
which represents a polynomial with n
+ 1 terms, |
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or written in the form of the sum formula
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called the binomial theorem. |
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binomial coefficients can also be
obtained by using Pascal's triangle. |
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triangular array of integers, with 1 at the
apex, in which each number is the sum of the two
numbers above it in the preceding row, as is
shown in the initial segment in the diagram, is
called Pascal's triangle. |
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| So,
for example the last row of the triangle
contains the sequence of the coefficients of a
binomial of the 5th power. |
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| n |
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1 |
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1 |
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1 |
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| 2 |
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1 |
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2 |
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1 |
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1 |
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3 |
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1 |
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4 |
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6 |
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1 |
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1 |
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5 |
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10 |
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10 |
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5 |
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1 |
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1 |
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| The difference of two
squares, multiplying |
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(a
- b)
· (a + b)
= a2 -
ab +
ab -
b2
= a2
- b2 |
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| Examples: |
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a) (x
- 2y) · (x
+ 2y)
= x2 - 2xy
+ 2xy
- 4y2
= x2 - 4y2 |
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b) (3a
+ 1) · (3a
- 1)
= 9a2 +
3a -
3a -
1
= 9a2 - 1 |
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| The difference of two
squares, factoring |
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a2
- b2
= (a - b)
· (a + b) |
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| Examples: |
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a) 1
- 16y2
= 12
- (4y)2
= (1 - 4y)
· (1 + 4y) |
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b) 1/9a4
- 0.0001
= (1/3a2)2 -
(0.01)2
= (1/3a2 - 0.01) · (1/3a2
+ 0.01) |
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| The difference of two
cubes |
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a3
- b3
= (a - b)
· (a2
+ ab
+ b2) |
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| Examples: |
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a) 8x3
- 125
= (2x)3 - 53
= (2x - 5) · (4x2
+ 10x
+
25) |
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b) 1
- 27a3
= 13 - (3a)3
= (1 - 3a) · (1
+ 3a
+ 9a2) |
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| The sum of two cubes |
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a3
+
b3 =
(a +
b) · (a2
-
ab +
b2) |
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| Examples: |
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a) 8
+ x3
= 23 + x3
= (2 + x) · (4
- 2x
+ x2) |
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b) 64a3
+ 0.001
= (4a)3 +
0.13
= (4a + 0.1) · (16a2
- 0.4a
+ 0.01) |
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Functions
contents A
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
