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| Algebraic
Expressions |
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Terms, variables, constants -
coefficients
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Monomial, binomial, trinomial, . . . , polynomial
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Simplifying algebraic
expressions |
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Like terms, collecting
or combining like terms |
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Evaluating 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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Factoring algebraic
expressions, methods |
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Finding (determining) a common
factor |
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Grouping
like terms, grouping and factoring four terms |
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The square of a
binomial - perfect square trinomials |
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Difference of two squares |
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Factoring quadratic trinomials |
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Sum and difference of cubes |
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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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| Terms, variables, constants -
coefficients
|
| An
algebraic expression is one or more algebraic terms
containing variables and constants connected by |
| mathematical
operations. |
| Terms
are the elements separated by the plus or minus signs. |
| In
algebraic expressions, variables are letters, such as a,
b, c,
or x,
y,
z,
that can have different values. |
| Constants
are the terms or elements represented only by numbers. |
| Coefficients
are the number part of the terms that multiply a variable or
powers of a variable. |
|
| Example: |
|
a2
- 6a
+ 5 -
is
the algebraic
expression of three terms, where, a
is
the variable, -
6 is |
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|
the
coefficient of a
of the middle
term,
and
5 is the constant. |
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| Monomial, binomial, trinomial, . . . polynomial
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| An
algebraic expression consisting of a single term is called a
monomial, expression consisting of two terms |
| is binomial, three
terms trinomial and an expression with more than three terms is
called polynomial. |
| A
monomial is a term containing variables with only nonnegative
integers as exponents. |
|
| Examples: |
|
-3x2
and
2ab
are monomials, a3
- b3
is
the binomial, |
|
|
x2
-
4x
+ 4 is
the trinomial, and
a3
- 3a2
+
3a
- 1
is the polynomial.
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| Simplifying algebraic
expressions |
| By
simplifying an algebraic expression, we mean reducing it in the
simplest possible form which mainly |
| involves: multiplication
and division, removing (expanding) brackets and collecting (adding
and subtracting) |
| like
terms. |
| Like
terms are those terms which contain the same powers of same
variables and which can only differ in |
| coefficients. |
|
| Examples: |
|
a)
- 4a3
+
3a2
+
5a3
- 7a2
= (- 4
+
5)
· a3
+
(3 - 7)
· a2 = a3 - 4a2, |
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|
b)
(x2
- x
+
1)
·
(x
+
1) =
x3 - x2
+
x
+
x2
- x
+
1 = x3
+
1. |
|
|
| Evaluating algebraic expressions |
| To
evaluate an algebraic expression means to replace (substitute)
the variables in the expression with |
| numeric values that are
assigned to them and perform the operations in the expression. |
|
| Example: |
|
Evaluate
the expression x2
- 6xy
+
9y2
for x
= 2 and y
= -1. |
| Solution: |
|
22 - 6
·
2
·
(- 1)
+
9
·
(-1)2
= 4 +
12 +
9 = 25. |
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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. |
|
| Examples: |
|
a)
(a
-
b)2 = (a
-
b)
· (a
-
b) =
a2
-
ab
-
ab
+
b2
= a2
-
2ab
+
b2, |
|
|
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. |
|
|
| 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)
= |
|
|
= 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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| Factoring algebraic
expressions |
| Factoring
algebraic expression by finding (determining) a common factor |
| Examples: |
|
a) 3x
-
6y
= 3 · (x
-
2y),
b) xy
- y2
= y ·
(x - y),
c) a
-
a2
= a ·
(1 -
a), |
|
|
d) x3
-3x2
+
x
= x ·
(x2
- 3x
+1),
e) x(a
+
b)
- (a
+
b)
= (a
+
b)
· (x -
1), |
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|
f)
a(x
- 3y)
- x
+
3y =
a(x -
3y) - (x
- 3y)
= (x - 3y)
·
(a
- 1). |
|
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| Grouping
like terms, grouping and factorizing four terms |
| An addition
sign, or plus sign, in front of the brackets leaves the sign of
every term inside the brackets |
| unchanged. |
| A minus sign
in front of the bracket indicates that, when removing the
bracket, the sign of all terms inside |
| must be changed. |
| Examples: |
|
a) ax
- bx - a
+
b
= x(a
-
b)
- (a
-
b)
= (a -
b)
·
(x
-
1), |
|
|
b) a
- 1
- ab
+
b
= (a
- 1)
- b
·
(a
- 1)
= (a
- 1)
· (1 -
b), |
|
|
c) x2
+
ax
- bx
- ab
= x(x
+
a)
- b
·
(x
+
a)
= (x +
a)
·
(x
-
b), |
|
|
d)
5ab2
- 3a3
- 10b3
+ 6a2b
= 5b2(a -
2b)
-3a2(a
- 2b)
= (a - 2b)(5b2
- 3a2). |
|
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| The
square of a binomial - perfect squares trinomials |
| Examples: |
|
a) 1
- 4x
+ 4x2
= 12 -
2 ·
2x
+ (2x)2
= (1 - 2x)2
= (1 - 2x)
· (1 -
2x), |
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b) a5
+ 6a4b
+ 9a3b2
= a3 ·
(a2
+ 6ab
+ 9b2
) = a3(a
+ 3b)2
= a3(a
+ 3b)(a
+ 3b). |
|
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| Difference of
two
squares |
| Examples: |
|
a) 16x2
- 1
= (4x)2
- 12
= (4x
-1)
· (4x
+1), |
|
|
b) 5y3
- 20x2y
= 5y
· (y2
- 4x2)
= 5y
[y2
- (2x)2]
= 5y(y
- 2x)(y
+ 2x), |
|
c)
9x2
-
(x
+ 2)2
= [3x -
(x
+ 2)]
· [3x +
(x
+ 2)]
= (2x
-2)
·
(4x
+ 2)
= 4(x
-1)
· (2x
+1). |
|
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| Factoring
quadratic trinomials |
| A
quadratic trinomial ax2
+ bx
+ c
can be factorized as |
| ax2
+ bx
+ c
= a·[x2
+ (b/a)·x
+ c/a]
= a·(x
-
x1)(x
-
x2), |
where
x1
+
x2
= b/a and
x1·
x2
= c/a |
|
| That
means, to factor a quadratic trinomial we should find such a
pair of numbers x1
and x2
whose sum |
|
equals b/a
and whose product equals c/a. |
| Therefore,
when a >
0 and the constant term c
is negative, then the signs of x1
and x2
will be different but |
| when c
is positive, their signs will be the same. |
|
|
Examples:
|
|
a) x2
- 3x
-10
= x2
+ (-5
+
2)·x
+ (-5)·(+2)
= x2
- 5x
+
2x
-10
= |
|
|
|
= x ·
(x
- 5)
+ 2 ·
(x
- 5)
= (x
- 5)
·
(x
+ 2), |
|
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|
|
|
b) 2x2
- 7x
+ 3
= 2 ·
(x2
- 7/2x
+ 3/2)
= 2(x2
- 1/2x
- 3x
+ 3/2)
= |
|
|
= 2[x·(x
- 1/2)
- 3·
(x - 1/2)]
= 2·
(x - 1/2)·(x
- 3)
= (2x
- 1)·
(x - 3), |
|
|
|
|
|
c) 3x2
- x
- 2
= 3(x2
- 1/3x
- 2/3)
= 3(x2 +
2/3x
- x
- 2/3)
= |
|
|
= 3[x·(x
+ 2/3)
-
(x + 2/3)]
= 3·(x
+ 2/3)·(x
- 1)
= (3x + 2)·(x
- 1). |
|
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|
Sum and difference of cubes |
|
Examples:
|
|
a) x3
+ 8
= x3 + 23
= (x + 2)
·
(x2
- 2x
+ 22), |
|
|
|
since (x + 2)·(x2
- 2x
+ 4)
= x3
- 2x2
+ 4x
+ 2x2
- 4x
+ 8
= x3
+ 8, |
|
|
|
|
|
b) 8a3
-125
=
(2a)3 -
53
= (2a - 5)·
[(2a)2
+
(2a)·5 + 52]
= (2a - 5)(4a2
+ 10a
+ 25), |
|
|
since (2a
- 5)(4a2
+ 10a
+ 25)
= 8a3 +
20a2
+ 50a
- 20a2
- 50a
-125
= 8a3 -125. |
|
|
| Using
a variety of methods including combinations of the above to
factorize expressions |
|
Examples:
|
|
a) x2
- 2xy
+ y2
+ 2y
- 2x
=
(x
- y)2
- 2(x
- y)
=
(x
- y)(x
- y
- 2), |
|
|
|
|
|
|
b) x2
- y2
+ xz
- yz
=
(x
- y)(x
+ y)
+ z(x
- y)
= (x
- y)(x
+ y
+ z), |
|
|
|
|
|
c) 4x2
- 4xy
+ y2
- z2
= (2x
- y)2
- z2
=
(2x
- y
- z)(2x
- y
+ z),
|
|
|
|
| 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). |
|
|
| Factoring and
expanding algebraic
expressions, rules
for transforming algebraic
expressions |
| Expanding algebraic
expressions |
|
The
square of a binomial, a perfect square trinomial |
|
(a +
b)2 = a2
+ 2ab
+
b2, |
|
|
(a
- b)2 =
a2
- 2ab
+
b2, |
|
|
The
square of a trinomial |
|
(a - b
+
c)2 = a2
+ b2
+ c2
- 2ab
+ 2ac
- 2bc, |
|
|
The
cube of a binomial |
|
(a + b)3
= a3 + 3a2b
+
3ab2
+ b3, |
|
|
(a - b)3
= a3
- 3a2b
+
3ab2
- b3, |
|
| Factoring algebraic
expressions |
|
Difference of
two
squares |
|
|
x2
- y2
= (x - y)
·
(x
+
y), |
|
|
Sum and difference of cubes |
|
x3 - y3
= (x - y)
·
(x2
+ xy
+ y2), |
|
|
x3 +
y3
= (x + y)
·
(x2 - xy
+ y2). |
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| Beginning
Algebra Contents |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
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