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ALGEBRA
- solved problems
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Algebraic
expressions
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Simplifying algebraic
expressions |
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1. |
Simplify algebraic expressions.
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Solutions: |
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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. |
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Evaluating algebraic expressions |
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2. |
Evaluate
the expression x2
- 6xy
+
9y2
for x
= 2 and y
= -1.
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Solution: |
x2
- 6xy
+
9y2
=
22 - 6
· 2
· (- 1)
+
9 ·
(-1)2
= 4 +
12 +
9 = 25. |
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Expanding
algebraic expression by removing parentheses ( brackets) |
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3. |
Expand given expressions.
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Solutions: |
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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)
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4. |
Square given binomials.
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Solutions: |
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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) |
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5. |
Square given trinomials.
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Solutions:
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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 |
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6. |
Cube (rise to third power) given binomials.
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Solutions:
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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 |
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7. |
Factorize given expressions.
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Solutions:
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a) 3x
- 6y
= 3 · (x
- 2y),
b) xy
- y2
= y ·
(x - y),
c) a
-
a2
= a · (1 -
a), |
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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
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8. |
Factorize given expressions.
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Solutions:
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a) ax
- bx
- a
+
b
= x(a
-
b)
- (a
-
b)
= (a
-
b)
·
(x
-
1), |
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b) a
- 1
- ab
+
b
= (a
- 1)
- b
·
(a
- 1)
= (a
- 1)
· (1 -
b), |
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c) x2
+
ax
- bx
- ab
= x(x
+
a)
- b
·
(x
+
a)
= (x +
a)
· (x
-
b), |
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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
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9. |
Factorize given expressions.
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Solutions:
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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 |
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10. |
Factorize given expressions.
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Solutions:
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a) 16x2
- 1
= (4x)2
- 12
= (4x
-1)
· (4x
+1), |
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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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Solved
problems contents |
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