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Algebraic
Expressions |
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Sum and difference of cubes |
Using
a variety of methods including combinations of the above to
factorize expressions |
Factoring and
expanding algebraic
expressions, rules
for transforming algebraic
expressions |
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Sum and difference of cubes |
Examples:
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a) x3
+ 8
= x3 + 23
= (x + 2)
·
(x2
- 2x
+ 22), |
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since (x + 2)·(x2
- 2x
+ 4)
= x3
- 2x2
+ 4x
+ 2x2
- 4x
+ 8
= x3
+ 8, |
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b) 8a3
-125
=
(2a)3 -
53
= (2a - 5)·
[(2a)2
+
(2a)·5 + 52]
= (2a - 5)(4a2
+ 10a
+ 25), |
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since (2a
- 5)(4a2
+ 10a
+ 25)
= 8a3 +
20a2
+ 50a
- 20a2
- 50a
-125
= 8a3 -125. |
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Using
a variety of methods including combinations of the above to
factorize expressions |
Examples:
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a) x2
- 2xy
+ y2
+ 2y
- 2x
=
(x
- y)2
- 2(x
- y)
=
(x
- y)(x
- y
- 2), |
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b) x2
- y2
+ xz
- yz
=
(x
- y)(x
+ y)
+ z(x
- y)
= (x
- y)(x
+ y
+ z), |
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c) 4x2
- 4xy
+ y2
- z2
= (2x - y)2
- z2
=
(2x
- y
- z)(2x
- y
+ z), |
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d) a3
- 7a
+ 6
= a3 - a
- 6a
+ 6
= a(a2 -1)
- 6(a
-1)
= (a -1)·[a(a
+ 1)
- 6]
= (a -1)(a2
+ a
- 6)
= |
= (a -1)(a2
+ 3a
- 2a
- 6)
= (a -1)[a(a
+ 3)
- 2(a
+ 3)]
= (a -1)(a
+ 3)(a
- 2). |
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Factoring and
expanding algebraic
expressions, rules
for transforming algebraic
expressions |
Expanding algebraic
expressions |
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The
square of a binomial, a perfect square trinomial |
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(a +
b)2 = a2
+ 2ab
+
b2, |
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(a
- b)2 =
a2
- 2ab
+
b2, |
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The
square of a trinomial |
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(a - b
+
c)2 = a2
+ b2
+ c2
- 2ab
+ 2ac
- 2bc, |
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The
cube of a binomial |
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(a + b)3
= a3 + 3a2b
+
3ab2
+ b3, |
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(a - b)3
= a3
- 3a2b
+
3ab2
- b3, |
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Factoring algebraic
expressions |
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Difference of
two
squares |
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x2
- y2
= (x - y)
·
(x
+
y), |
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Sum and difference of cubes |
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x3 - y3
= (x - y)
·
(x2
+ xy
+ y2), |
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x3 +
y3
= (x + y)
·
(x2 - xy
+ y2). |
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Beginning
Algebra Contents B |
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