Remainder and Factor Theorem

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Remainder and Factor Theorem

• Polynomials
• Combining polynomials
• Function notation
• Division of Polynomial
• Remainder Theorem
• Factor Theorem

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Polynomials

• An expression that can be written in the form
• a bx cx2 dx3 ex4 .
• Things with Surds (e.g. ?x 4x 1 ) and
  reciprocals (e.g. 1/x x) are not polynomials
• The degree is the highest index
• e.g 4x5 13x3 27x is of degree 5

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Polynomials can be combined to give new
polynomials
2x4 6x2 -3x -3
6x6
-10x5
-6x4
12x4
-20x3
-12x2
6x6
-10x5
12x4
-6x4
-20x3
-12x2
Gather like terms
6x6 - 10x5 6x4 - 20x3 -12x2
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Function Notation

• An polynomials function can be written as
• f(x) a bx cx2 dx3 ex4 .
• f(x) means function of x
• instead of y .
• e.g f(x) 4x5 13x3 27x
• f(3) means .
• the value of the function when x3
• e.g. for f(x) 4x5 13x3 27x
• f(3) 4 x 35 13 x 33 27 x 3
• f(3) 972 351 81 1404

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Combining Functions (1)

• Suppose
• f(x) x3 2x 1
• g(x) 3x2 - x - 2
• g(x) or p(x) or q(x) or ..
• Can all be used to define different functions

• We can define a new function by any linear or
  multiplicative combination of these
• e.g. 2f(x) 3g(x)
• 2(x3 2x 1) 3(3x2 - x - 2)
• e.g. 3 f(x) g(x)
• 3(x3 2x 1)(3x2 - x - 2)

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Combining Functions (2)

• We can define a new function by any linear or
  multiplicative combination of these
• e.g. 2f(x) 3g(x)
• 2(x3 2x 1) 3(3x2 - x - 2)
• 2x3 4x 2 9x2 -3x -6

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2

- 4
2x3 9x2 x - 4
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Combining Functions (3)
e.g. 3 f(x) g(x) 3(x3 2x 1)(3x2 - x - 2)
Do multiplication table gather like terms and
then multiply through by 3
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Finding one bracket given the other
Fill in the empty bracket
x2 - x - 20 (x 4)( )
x2 - x - 20 (x 4)(x )
x2 - x - 20 (x 4)(x - 5)
Expand it to check (x 4)(x - 5) x2
4x - 5x -20 x2 - x - 20
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We can do division now
f(x) (x2 - x - 20) ? (x 4)
(x 4) x f(x) (x2 - x - 20)
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Finding one bracket given the other - cubics
x3 3x2 - 12x 4 (x - 2)(x2 )
x3 3x2 - 12x 4 (x - 2)(x2 -2)
x3 3x2 - 12x 4 (x - 2)(x2 5x -2)
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Finding one bracket given the other - cubics
(2.1)
Using a multiplication table
x3
3x2
-12x
4
x3
4
Can put x3 and 4 in. They can only come from
1 place. So a 1, c -2
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Finding one bracket given the other - cubics
(2.2)
Using a multiplication table
3x2
-12x
-2x
x3
-2x2
4
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Finding one bracket given the other - cubics
(2.3)
5x
Using a multiplication table
3x2
-12x
bx2
-2x
x3
-2x2
-2bx
4
Gather like terms
Either way, b 5
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Dealing with remainders
remainder
Using a multiplication table
x3
Can put x3 This can only come from 1 place. So a
1
15
Dealing with remainders
x2
remainder
Using a multiplication table
x3
-x2
x
x3
15
bx2 2x2 -x2
b 2 -1
b -3
16
Dealing with remainders
x2 -3x
remainder
Using a multiplication table
x3
-x2
x
x3
15
-3x2
2x2
cx - 6x x
c - 6 1
c 7
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Dealing with remainders
x2 3x 7
remainder
Using a multiplication table
x3
-x2
x
7x
x3
15
-3x2
-6x
2x2
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Dealing with remainders
x2 3x 7
remainder
Using a multiplication table
x3
-x2
x
7x
x3
15
-3x2
14
-6x
2x2
Remainder
The numerical term (15) comes from the 14 and
the remainder R
15 14 R
So, R 1
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Dealing with remainders
x2 3x 7
Using a multiplication table
x3
-x2
x
7x
x3
15
-3x2
14
-6x
2x2
Remainder
The numerical term (15) comes from the 14 and
the remainder R
15 14 R
So, R 1
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Division with Remainders
What is f(x) divided by x2 . and what is
the remainder
This is exactly the same as
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What is f(x) divided by x2? . and what is
the remainder?
What is f(x) divided by x2? (x2 3x 7)
. and what is the remainder? 1
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The Remainder Theorem example 1
x2 3x 7
What is f(x) divided by x2? (x2 3x 7)
. and what is the remainder? 1
If we calculate f(-2) ..
f(-2) (-2)3 (-2)2 -2 15 -8 -4 - 2
15 1
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The Remainder Theorem example 2
x2 4x - 1
What is p(x) divided by x-2? (x2 4x - 1)
. and what is the remainder? 8
If we calculate p(2) ..
p(2) (2)3 2(2)2 - 9(2) 10 8 8 - 18
10 8
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The Remainder Theorem
When p(x) is divided by (x-a) . the remainder
is p(a)
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The Factor Theorem example
For bigger values of x the x3 term will
dominate and make p(x) larger
What value of p(...)0, hence will give no
remainder?
If we calculate p(0) 2(0)3 5(0)2 0 - 12
-12
p(1) 2(1)3 5(1)2 1 - 12 2-51-12 -14
p(2) 2(2)3 5(2)2 2 - 12 16-202-12 -16
p(3) 2(3)3 5(3)2 3 - 12 54-453-12 0
By the Remainder Theorem - the factor (x-3)
gives no remainder
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The Factor Theorem example
p(3) 2(3)3 5(3)2 3 - 12 54-453-12 0
By the Remainder Theorem - the factor (x-3)
gives no remainder
So (x-3) divides exactly into p(x) (x-3)
is a factor
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The Factor Theorem
For a given polynomial p(x) If p(a) 0 then
(x-a) is a factor of p(x)
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When is p(x) divided by x2 the remainder is 5
Which theorem?
The Remainder Theorem
If we calculate p(-2) ..
p(-2) (-2)3 b(-2)2 b(-2) 5 -8 4b
- 2b 5 2b - 3
By the Remainder theorem 2b - 3 5
2b 8 b 4
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a) Find f(2)
f(2) (2)3 3(2)2 - 6(2) - 8 8 12 - 12
- 8 0
b) Use the Factor Theorem to write a factor of
f(x)
For a given polynomial p(x) If p(a) 0 then
(x-a) is a factor of p(x)
f(2) 0 . so (x-2) is a factor of x3 3x2 - 6x
- 8
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c) Express f(x) as a product of 3 linear factors
.. means (x-a)(x-b)(x-c)x3 3x2 - 6x - 8
We know (x-2)(x-b)(x-c)x3 3x2 - 6x - 8
. consider (x-2)(ax2bxc)x3 3x2 - 6x - 8
a?
a1 so x x ax2 x3
(x-2)(x2bxc)x3 3x2 - 6x - 8
c4 so -2 x 4 -8
c?
(x-2)(x2bx4)x3 3x2 - 6x - 8
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c) Express f(x) as a product of 3 linear factors
(x-2)(x2bx4)x3 3x2 - 6x - 8
Expand need only check the x2 or x terms
bx2 -2x2 3x2 .
EASIER
b - 2 3 b5
Or - 2bx 4x - 6x .
-2b 4 -6 b5
HARD
(x-2)(x25x4)x3 3x2 - 6x - 8
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c) Express f(x) as a product of 3 linear factors
(x-2)(x25x4)x3 3x2 - 6x - 8
(x25x4) (x4)(x1)
So,
(x-2)(x4)(x1) x3 3x2 - 6x - 8
. a product of 3 linear factors
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(x-2)(x4)(x1) x3 3x2 - 6x - 8
. Sketch x3 3x2 - 6x - 8
y x3 3x2 - 6x - 8
y (x-2)(x4)(x1)
Where does it cross the x-axis (y0) ?
(x-2)(x4)(x1) 0
Either (x-2) 0 x2
Or (x4) 0 x-4
Or (x1) 0 x-1
Where does it cross the y-axis (x0) ?
y (0)3 3(0)2 - 6(0) - 8 -8
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Factor and Remainder Theorem
Either (x-2) 0 x2
Where does it cross the x-axis (y0) ?
Or (x4) 0 x-4
Or (x1) 0 x-1
y -8
Where does it cross the y-axis (x0) ?
Goes through these -sketch a nice curve