Download Polynomials

A monomial, in one variable, is the
product of a constant times a variable
raised to a nonnegative integer power.
Thus, a monomial is of the form:
ax
k
where a is a coefficient, x is a variable,
and k > 0 is an integer.
Monomial
3x
4
2x
9
Coefficient
3
Degree
4
2
1
-9
0
A Binomial in one or more variable is
the combination of two monomials
separated by addition or subtraction
and raised to nonnegative integer
powers. Thus, a binomial is of the
form
ax  bx
k
n
where a is a constant, x is a variable,
and k > 0 is an integer.
Binomial
Coefficients
3x  9 x
4
2 x  8x
3  5z
2
Degree
3, 9
4
2 ,8
2
-9, -5
1
A Trinomial in one or more variable is
the combination of three monomials
separated by addition or subtraction
and raised to nonnegative integer
powers. Thus, a binomial is of the
form
ax  bx  cx
k
n
m
where a is a constant, x is a variable,
and k > 0 is an integer.
Trinomial
3x  9 x  5 x
4
Coefficients Degree
2
2 x  8x  1
2
3  5z  6 z
3
3, 9, 5
2 ,8,1
3, -5, 6
4
2
3
A polynomial in one variable is an
algebraic expression of the form:
an x n  an 1 x n 1 
 a1 x  a0
Where an , an 1..., a1 , a0 are constants, called
Coefficients of the Polynomial, n  0 is an
integer,and x is a variable.
If an  0, it is called the leading Coefficient,
and is called the Degree of the Polynomial.
n
Example:
Polynomial:
Degree:
2 x  3x  x  5
4
2
4
Coefficients:
2, -3, 1, -5
Leading Coefficient:
2
x  2 x  x  3x
2
3
Standard Form:
(Powers in Descending Order)
Degree: (Highest Power)
4
3x4-2x3+x2+x
4
Terms:
(Monomials of the Polynomial) x2, x, -2x3, 3z4
Coefficients:
(Real Numbers in front of a term)
1, -2, 3
Leading Coefficient:
(Coefficient of the Highest Power)
3
Turning Points:
(The Highest Power minus one)
3
A Zero Polynomial:
(All zero Coefficients)
0x40x3+0x2+0x
Addition of Polynomials
Combining “like” termsMeaning they must have the same variables all with the
same powers/exponents
3
3
2
(

3
x

2
x

4
)

(
4
x

3
x
 2)
Add:
You can add by inspection of like terms:
(3x 3  2 x  4)  (4 x 3  3x 2  2) 
x 3  3x 2  2 x  2
You can group the “like” terms:
(3x 3  4 x 3 )  (3x 2 )  (2 x)  (2  4) 
x 3  3x 2  2 x  2
Line up “like” terms and adding as you would regular numbers:
 3x 3
 4 x 3  3x 2
 2x  4
2
x 3  3x 2  2 x  2
Subtraction of Polynomials
Subtract: (5x 2  3x  6)  (9 x 2  5x  3)
Adding the Opposite- Changing Subtraction to Addition
(The same as multiplying it by a Negative one)
(5x2-3x+6)+(-9x2+5x+3)
You can add by inspection of like terms:
(5 x 2  3x  6)  (9 x 2  5 x  3) 
- 4x 2  2x  9
You can now ADD by grouping the “like” terms:
(5 x  9 x )  (3x  5 x)  (6  3) 
2
2
 4x  2x  9
2
Line up “like” terms and ADD as you would regular numbers:
5 x 2  3x  6
2
 (9 x  5 x  3)
 4x 2  2x  9
Multiplication of Polynomials
To Multiply a monomial times a polynomial
use the Distributive Property of Real Numbers
Multiply:
2 x ( x  7 x  10 x  4)
2
3
2
2 x 2 ( x 3  7 x 2  10 x  4) 
2 x 2  3  14 x 2  2  20 x1  2  8 x 2 
(2 x 5  14 x 4  20 x 3  8x 2 )
To Multiply Two Binomials use the FOIL Method:
First, Outer, Inner, Last
Multiply:
(2 x 3  5)  ( x 2  4)
(2x3)(x2) + (2x3)(-4) + (5)(x2) + (5)(-4) =
2 x 5  5 x 2  8 x 3  20
To Multiply any two Polynomial s use the
Distributive Property of Real Numbers
and combine the “like” terms.
Multiply: ( x 2  2 x)  (4 x 3  2 x 2  3x)
(x2)(4x3)+(x2)(-2x2)+(x2)(3x)+(2x)(4x3)+(2x)(-2x2)+(2x)(3x)
4 x 5  2 x 4  3x 3  8 x 4  4 x 3  6 x 2 
4 x5  4 x 4  x3  6 x 2
To Multiply any two Polynomial s use the Method
For Regular Vertical Multiplication Method with
Real Numbers
Multiply: ( x 2  2 x)  (4 x 3  2 x 2  3x)
4 x 3  2 x 2  3x
x2  2x
8x4 - 4x3 + 6x2
4x5 -2x4 +3x3
4x5+ 6x4 – x3 + 6x2
Division of Polynomials
When dividing by a Monomial Separate terms
over the common denominator and and simplify
using Exponent Rules:
10a 5 b 4  2a 3b 2  6a 2 b
2a 2 b
Divide:
Split into separate terms with the denominator
5
4
3
2
2
10a b
2a b
6a b


2
2
2a b
2a b
2a 2 b
10 5 4  2 1 2 3 2  2 1 6 2  2 1
a b a b  a b a b  a ba b
2
2
2
5a 5  2 b 4 1  1a 3  2 b 2 1  3a 2  2 b1 1
5a 3b 3  a1b1  3a 0 b 0
5a 3b 3  ab  3
Division of Polynomials
When dividing by another Polynomial
Find the Quotient and
remainder when
2 x  8x  x  4
3
2
is divided by
x 4
2 x2  1 2 x3  8x2  x  4
2x2  1
using Long Division:
x
2 x3
 8x  2 x
2
 8x2
Check:
2x
2

 1 x  4  2 x  8 
4
2x  8
 2 x  8x  x  4  2 x  8
3
2
 2 x  8x  x  4
3
2
2 x  8x  x  4
2x  8
Thus,
 x4 2
2
2x  1
2x  1
3
2
1234-
Synthetic Division:
ONLY use when power of divisor is one
Write all Coefficients in order of descending
Power, if a power is missing write in a zero.
Multiply on the diagonals and Add the Columns
as you move across the grid.
Remember each time you divide by a number
the power of the original Polynomial goes down
one power.
Divide:
3x  4 x  x  7
3
2
by
x 1
Divisor is x + 1 and the Power is one.
x + 1= 0 which gives x = -1
-1
3
4
-3
1
-1
7
0
3
1
0
7
Polynomial now goes down a Power
3x  4 x  x  7

x 1
3
2
7
3x  x 
x 1
2