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Algebraic
Expressions |
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Expanding
algebraic expression by removing parentheses (i.e. brackets) |
The square of a
binomial – the perfect square trinomial |
The square of a trinomial |
The cube of binomial |
The binomial expansion
algorithm |
The difference of two
squares, multiplying |
The difference of two
squares, factoring |
The difference of two
cubes |
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. |
Some
important binomial products like perfect squares, and difference
of two squares are used to help with factoring algebraic
expressions. |
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Examples: |
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a)
(a
- b)2
= (a
- b)
· (a
- b) =
a2
- ab
- ab
+
b2
= a2
- 2ab
+
b2, |
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b)
(a
- b)
·
(a
+
b) =
a2 - ab
+
ab
- b2 =
a2 -
b2, |
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c)
(x
+
y)
· (x2
- xy
+
y2) =
x3 - x2y
+
xy2
+
x2y
- xy2
+
y3
= x3
+
y3. |
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The
square of a binomial (or binomial square ) |
To the
square of the first term add twice the product of the two terms
and the square of the last term. |
Examples: |
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a)
(a +
b)2
= (a
+ b)
·
(a
+ b) =
a2 +
ab
+ ab
+
b2
= a2
+ 2ab
+
b2, |
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b)
(2x +
3)2 =
(2x)2
+ 2
· (2x) ·
3
+
32
= 4x2 +
12x +
9, |
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c)
(x
- 2y)2 =
x2 +
2 · x
·
(-2y)
+
(-2y)2 =
x2 -
4xy +
4y2. |
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Squaring
trinomial (or trinomial square) |
To the sum of squares of the 1st, the 2nd and the 3rd term add, twice the product of the 1st and the 2nd term, twice the product of the 1st and the 3rd term, and twice the product of the 2nd and the 3rd term. |
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Examples:
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a) (x2
- 2x
+ 5)2
= (x2)2
+ (2x)2
+ 52
+ 2
·
x2 ·
(-2x)
+ 2
·
x2 ·
5 +
2 ·
(-2x)
·
5
= |
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= x4
+ 4x2
+ 25
- 4x3
+ 10x2
- 20x
= x4 - 4x3
+ 14x2
- 20x
+ 25, |
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b) (a3
-
a2b - 3ab2)2
= (a3)2 + (a2b)2
+ (3ab2)2
+ 2a3
(-a2b)
+ 2a3
(-3ab2)
+ 2(-a2b)
(-3ab2)
= |
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= a6
+ a4b2
+ 9a2b4
- 2a5b
- 6a4b2
+ 6a3b3
= a6 - 5a4b2
+ 9a2b4
- 2a5b
+ 6a3b3. |
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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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is
called the binomial theorem. |
The
binomial coefficients can also be
obtained by using Pascal's triangle. |
The
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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3 |
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4 |
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6 |
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5 |
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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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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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Intermediate
algebra contents |
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