polynomial identities division
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Polynomials and Partial Fractions
Objectives
In this lesson, you will learn how to perform arithmetic operations on polynomials.
4.1 Polynomials
2P 5 3 1x x x
The following expressions are polynomials.
3 23 4 2 1x x x
4 2127 3x x We also use this
form of function notation to denote
a polynomial.7 is the coefficient of x4 and
½ is the coefficient of x2.
A polynomial in a variable x, is a sum of terms, each of the form axn, where a is a constant and n is a non-negative integer.
Polynomials and Partial Fractions
2P 3 3 3 1
2 21 4 6 4x x x x
2 2P 2Q 1 2 2 3 2x x x x x x
25 5 5x x
Substitute for x in P(x).
Combine the two polynomial functions.
If P(x) = x2 + x + 1 and Q(x) = 2x2 – 3x + 2, find
Polynomials and Partial Fractions
Example
(a) P(3),(b) P(x) + 2Q(x).
13
25 1x x
Polynomials and Partial Fractions
In this lesson, you will learn how to find unknown constants in a polynomial identity.
4.2 Identities
Objectives
2 22 4 4x x x
An expression involving polynomials that can be solved to find a specific value for x, is an equation.
2 2 4 3x x x 2 2 3 0x x
This is always true, so, it is an identity.
We have solved for x, so, this is an equation.
Equations and Identities
Polynomials and Partial Fractions
1 and 3x x
An expression involving polynomials that is true for all values of x is an identity.
3 22 6 5 2 2 2x x x x ax b
6 2 2a If x = 2, then
(x – 2) = 0.
The coefficients of x2 could
be used too.
Find the values of a and b in the following identity.
Polynomials and Partial Fractions
16 12 5 b 2 8a 3
2 2 6 2 5 b
9b
Let x = 2. Equate the coefficients of x.
4a
x-3 -2 -1 1 2 3
y
-4
-2
2
4
6
8
10
y = 2x3
– 6x + 5
y = (x – 2)(2x2
+ 4x + 2) + 9
Check the results
graphically.
Example
23 2 23 2 16 1 2 1 2x x x Ax x B x x C x
If x = 1, then(x – 1) = 0.
Find the values of A, B and C in the following identity.
Polynomials and Partial Fractions
18 3 6C C Let x = 1.
Let x = 2.
3 21 3 1 2 1 16 1 2C
If x = 2, then
(x – 2) = 0.
3 2 22 3 2 2 2 16 2 2 1 6 2 2A
8 12 4 16 4 24A 4 8 2A A
Equate the coefficients of x3.
1 A B 1B
Example
Polynomials and Partial Fractions
In this lesson, you will learn how to divide one polynomial by another.
4.3 Dividing Polynomials
Objectives
8 1327Subtract 1 × 8 from
13.
Divide 13 by 8.
A reminder about long division of integers.
Polynomials and Partial Fractions
1 8 1327
8
52
16 8 1327
48
47
1658 1327
40
7
8
5 48
4
Bring the 2 down.
Divide 52 by 8.
Subtract 6 × 8 from 52.
Bring the 7 down.
Divide 47 by 8.
Subtract 5 × 8 from 47.
divisordividend
quotient
remainder
Therefore 1327 8 165 7
We will now apply the same process to
polynomials.
For any division,
Polynomials and Partial Fractions
dividend = divisor × quotient + remainder
or
dividend ÷ divisor = quotient +divisor
remainder
4 3 21 3 2 7x x x x x Subtract x3 × (x – 1)
from x4 + 3x3.
Divide x4 by x.
This is the same method as long division with integers.
Polynomials and Partial Fractions
3
4 3 2
1 3 2 7
xx x x x x
4 3
3 2
4 2
x x
x x
3 2
4 3 2
4 1 3 2 7
x xx x x x x
3 2
2
4 4
2
x x
x x
3 2
4 3 2
4 2 1 3 2 7
x x xx x x x x
22 2
3
x x
x
4 3
3 4
x x
x
3 2
2
4 4
2
x x
x
Bring the – 2x2
down.
Divide 4x3 by x.
Subtract 4x2× (x – 1) from 4x3 –
2x2.
Bring the x down.
Divide 2x2 by x.
Subtract 2x× (x – 1) from 2x2 + x.
divisor dividend
quotient
remainder
22 2
3 7
x x
x
3 3
4
x
Divide 3x by x.
Subtract 3× (x – 1) from 3x –
7.
3 2
4 3 2
4 2 31 3 2 7
x x xx x x x x
Bring the – 7 down.
The following identity is always true
Polynomials and Partial Fractions
3 2
4 3 2
4 2 31 3 2 7
x x xx x x x x
dividend = divisor × quotient + remainder
+ remainderquotient
divisor dividend
4
4 3 2 3 23 2 7 1 4 2 3 4x x x x x x x x
Therefore
Example
Divide .3 22 4 2 by 2x x x x
3 22 2 4 2x x x x Subtract 2x2 × (x – 2) from 2x3 – 4x2.
Divide 2x3 by x.
Polynomials and Partial Fractions
3 22 4
0
x x
x
2
3 2
2 12 2 4 2
xx x x x
2
0
x
2
3 2
2 2 2 4 2
xx x x x
3 22 4
0
x x
Bring the x down.
Divide 0 by x.
There is no x term.
Bring the – 2 down.
Divide x by x.Subtract 1× (x – 2)
from x – 2.
2
3 2
2 2 2 4 2
xx x x x
3 22 4
2
x x
x
3 2 2Hence 2 4 2 2 2 1x x x x x
Example
Divide .4 3 2 24 5 2 by 2x x x x x
2 4 3 22 4 5 2x x x x x
Subtract 4x2 × (x2 – x + 2)
from 4x4 – 5x3 + x2
Divide 4x4 by x2.
Polynomials and Partial Fractions
4 3 2
3 2
4 4 8
7
x x x
x x
2
2 4 3 2
4 8
2 4 5 2
x x
x x x x x
28 8 16
6 14
x x
x
2
2 4 3 2
4
2 4 5 2
x x
x x x x x
4 3 2
3 2
4 4 8
7
x x x
x x
No term to bring down.
Divide –x3 by x2.
Subtract –x × (x2 – x + 2) from – x3 – 7x2.
Bring the – 2 down.
Subtract – 8 × (x2 – x + 2) from – 8 x2 +8x –
2.
2
2 4 3 2
4
2 4 5 2
x
x x x x x
3 2
2
2
8 2
x x x
x x
4 3 2 2 2Hence 4 5 2 2 4 8 6 14x x x x x x x x
3 2
2
2
8 2 2
x x x
x x
Divide –8x2 by x2.
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