|
| Polynomial |
|
|
|
|
Terms, the degree of a polynomial |
|
Addition and subtraction of
polynomials
|
|
Multiplication of polynomials
|
|
Division of polynomials
|
|
Factoring,
factoring by grouping, GCF (Greatest Common Factor) |
|
Factoring
trinomials and polynomials |
|
|
Rational
Expressions |
|
Simplification of rational expressions, reducing to lowest
terms
|
|
Multiplication and division of rational
expressions
|
|
Addition and subtraction of rational
expressions
|
|
Solving complex rational expressions
|
|
|
|
|
|
|
|
| Polynomial |
| Terms, the degree of a polynomial |
| The
polynomial is mathematical expression consisting of a sum of
terms each of which is a product of a |
| constant and one or more
variables raised to a non-negative integral power. If there is
only a single variable x, |
| the general
form is given by |
|
anxn + an-1xn-1 + an-2xn-2
+ . . . + a2x2
+ a1x + a0, |
| where
the ai
are real
numbers called coefficients and a0
is the constant term. |
| The
degree of a polynomial is the highest power or the sum of powers
in any term of a given polynomial or |
| algebraic expression. |
|
| Addition and subtraction of
polynomials
|
| We
add or subtract polynomials by combining their like
terms. The like terms are terms that have the same |
|
variables raised to the same exponents. |
|
| Example:
a)
( -
5x3 +
2x2
- x +
4) + ( -
4x2 +
3x - 7)
= - 5x3 +
(
2 -
4) · x2 +
(
-1
+
3) · x +
4 - 7
= |
|
= - 5x3 - 2x2 +
2x - 3 |
|
|
b) (
x4 -
3x3
+
5x - 1)
-
( - 2x4 +
x3
- 3x2 +
4)
= x4 -
3x3
+
5x - 1
+ 2x4 -
x3
+
3x2 -
4
= |
|
= (
1 +
2) · x4
+ (
-3
- 1)
· x3
+
3x2
+
5x - 5
= 3x4 -
4x3
+
3x2
+
5x - 5 |
|
| Multiplication of polynomials
|
| When
multiplying two polynomials together, multiply every term of one
polynomial by every term of the other |
| polynomial using
distributive property. |
|
| Example:
( -
2x3 +
5x2
- x +
1) ·
(
3x - 2)
= |
|
= 3x · (
- 2x3)
+ 3x · 5x2
+ 3x ·
(-x) + 3x · 1
+ (- 2) · ( -
2x3)
+ (- 2) · 5x2
+ (- 2) · ( -x)
+ (-
2) · 1
= |
|
= - 6x4
+ 15x3
- 3x2
+ 3x +
4x3 -
10x2
+ 2x -
2 = -
6x4
+ 19x3
-
13x2
+ 5x -
2 |
|
| Division of polynomials
|
| Divide
the highest degree term of dividend by the highest degree term
of the divisor to get the first term of the quotient. |
| Take the first term of the quotient and multiply it by every term of divisor.
Write this result below the dividend, making sure you line up all the terms with the terms of the dividend that has the same degree. |
| Subtract the result from the dividend, i.e., reverse all the signs of the terms of the result and add like terms. |
| Repeat the process of long division until the degree of the new obtained dividend is less than the degree of the divisor. |
|
|
| Note,
since each second line should be subtracted, the sign of each term is reversed. |
|
|
b)
(
3x4 +
x2 +
5) ¸
( x2 -
x -
1) = -
2x2 +
x -
1 |
|
|
|
|
 |
|
| |
|
| like |
17 ¸
5 = 3 +
2/5 |
| |
-15 |
| |
2 |
|
|
|
| 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) |
|
|
| Rational
Expressions |
|
|
Simplification of rational expressions, reducing to lowest
terms
|
|
Multiplication and division of rational
expressions
|
|
Addition and subtraction of rational
expressions
|
|
Solving complex rational expressions
|
|
| Simplification of rational expressions, reducing to lowest
terms
|
| A
rational expression is a fraction of which the numerator and the
denominator are polynomials. |
| A
rational expression is reduced to lowest terms if all common
factors from the numerator and denominator |
| are canceled. |
| To
reduce a rational expression to lowest terms first factorize
both the numerator and denominator as much |
| as possible then
cancel common factors, i.e., divide their numerator and
denominator by common factors. |
|
| Examples:
Reduce the following
rational expression to lowest terms. |
| |
 |
| |
 |
| |
 |
| |
factoring
out the minus sign, |
|
or |
|
|
| |
 |
| |
 |
| |
 |
|
|
| Multiplication and division of rational
expressions
|
| Use
the formulas for the multiplication
and the division of fractions. |
| |
 |
and |
 |
|
|
| Notice
the two special cases of which should be aware of: |
| |
 |
and |
 |
|
|
|
| Examples:
Perform the indicated
operations and reduce the answer to lowest terms. |
|
|
| Note,
since each second line should be subtracted, the sign of each term is reversed. |
|
| Addition and subtraction of rational
expressions
|
| To add or subtract two fractions with
the same denominator, add or subtract the numerators and write the |
| sum over the common denominator. |
| To add or subtract fractions with different
denominators: |
| First find
the least common denominator (lcd -the smallest number that can be
divided by each denominator). |
| Write equivalent fractions using this
denominator. Then add or subtract the fractions. |
| The
process for rational expressions is identical. |
|
| Examples:
Perform the indicated
operations and reduce the answer to lowest terms. |
|
|
|
| Solving complex rational expressions
|
| A complex or compound rational expression has fractional expressions in its numerator, denominator or both. |
| To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator |
| by the denominator. |
| Examples:
Simplify complex
fractions. |
|
|
|
|
|
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|
|
| Beginning
Algebra Contents |
|
 |
|
| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
