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| Algebraic
Expressions |
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Expanding
algebraic expression by removing parentheses (i.e. brackets) |
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The square of a
binomial (or binomial square ) |
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Cube of a binomial |
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The square of a
trinomial |
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| Expanding
algebraic expression by removing parentheses (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: |
|
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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| Beginning
Algebra Contents B |
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