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| Algebraic
Expressions |
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Expanding
algebraic expression by removing parentheses (i.e. brackets) |
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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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| Expanding
algebraic expression by removing parentheses (i.e. brackets) |
| The
operation of multiplying out algebraic expressions that involve
parentheses using the distributive property is often
described as expanding the brackets. |
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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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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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| 3 |
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1 |
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3 |
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3 |
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1 |
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| 4 |
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1 |
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4 |
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6 |
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4 |
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1 |
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| 5 |
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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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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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| Pre-calculus
contents A |
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
