|
| Factoring
Polynomials |
|
|
|
Factoring,
factoring by grouping, GCF (Greatest Common Factor) |
|
Factoring
trinomials and polynomials |
|
|
|
|
|
|
| Factoring,
factoring by grouping, GCF or GCD |
| Greatest Common
Factor (or Greatest Common Divisor or Highest Common Factor) |
| The
greatest common factor (GCF) is the greatest factor
that divides two given numbers. |
| The
greatest common factor of a polynomial is the largest monomial
that divides each term of the polynomial. |
|
| Examples:
a)
2a2x -
4a
=
2a
· (ax
-
2) |
|
b)
-2x3
+
4x2 -
10x
=
-2x
· (x2
- 2x
+ 5) |
|
c)
3a3 -
3a2b
+
6a2c
=
3a2
· (a
-
b
+ 2c) |
|
d)
-15a4b6 -
5a5b4
+
10a3b5
=
-5a3b4
· (3ab2
+
a2
-
2b) |
|
| Factoring
by grouping |
| Examples:
a)
x2 -
1 -
xy
+ y = (x
-
1)
· (x
+
1)
- y
· (x
-
1) =
(x
-
1)
· (x
-
y
+ 1) |
|
b) x3 -
5x2
- 3x+
15
= x2(x
-
5)
- 3(x
-
5) =
(x
-
5)
· (x2
- 3) |
|
| Factoring
trinomials
and polynomials |
| 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. |
|
| Examples:
a) x2 -
x
-
12
= x2 -
4x
+
3x
-
12
= x(x
-
4)
+
3(x
-
4)
=
(x
-
4)
· (x
+ 3) |
|
b) 2x2 -
5x
+
3
= 2(x2 -
5/2x
+
3/2)
= 2(x2 -
3/2x
- x
+
3/2)
=
2[x(x
- 3/2)
- (x
- 3/2)]
= |
|
= 2(x
- 3/2)(x
-
1)
=
(2x
-
3)(x
-
1) |
|
c) x3 -
3x
+
2
= x3 -
4x
+
x +
2
= x
(x2
-
4)
+
(x
+ 2)
=x(x
-
2)(x
+ 2)
+
(x
+ 2)
= |
|
= (x
+ 2)[x
(x
-
2)
+
1]
= (x
+ 2)(x2 -
2x+
1)
= (x
+ 2)(x
-
1)(x
-
1) |
|
|
or x3 -
3x
+
2
= x3 -
x
-
2x
+
2
= x
(x2
-
1)
-
2(x
-
1)
= x(x
-
1)(x
+ 1)
-
2
(x
-
1)
= |
|
= (x
-
1)[x(x
+ 1)
-
2] =
(x
-
1)(x2
+ x -
2)
=
(x
-
1)(x2
- x
+ 2x
-
2)
= |
|
= (x
-
1)[x(x
-
1)
+ 2(x
-
1)]
= (x
-
1)(x
-
1)(x
+ 2) |
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| Beginning
Algebra Contents C |
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| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
