Factorising polynomials

1
Factorising polynomials
This PowerPoint presentation demonstrates three
methods of factorising a polynomial when you know
one linear factor.
Click here to see factorising by inspection
Click here to see factorising using a table
Click here to see polynomial division
2
Factorising by inspection
If you divide x³ - x² - 4x 6 (cubic) by x 3
(linear), then the result must be quadratic.
Write the quadratic as ax² bx c.
x³ x² 4x - 6 (x 3)(ax² bx c)
3
Factorising by inspection
Imagine multiplying out the brackets. The only
way of getting a term in x³ is by multiplying x
by ax², giving ax³.
x³ x² 4x - 6 (x 3)(ax² bx c)
So a must be 1.
4
Factorising by inspection
Imagine multiplying out the brackets. The only
way of getting a term in x³ is by multiplying x
by ax², giving ax³.
x³ x² 4x - 6 (x 3)(1x² bx c)
So a must be 1.
5
Factorising by inspection
Now think about the constant term. You can only
get a constant term by multiplying 3 by c,
giving 3c.
x³ x² 4x - 6 (x 3)(x² bx c)
So c must be 2.
6
Factorising by inspection
Now think about the constant term. You can only
get a constant term by multiplying 3 by c,
giving 3c.
x³ x² 4x - 6 (x 3)(x² bx 2)
So c must be 2.
7
Factorising by inspection
Now think about the x² term. When you multiply
out the brackets, you get two x² terms.
-3 multiplied by x² gives 3x²
x³ x² 4x - 6 (x 3)(x² bx 2)
x multiplied by bx gives bx²
So 3x² bx² -1x²
therefore b must be 2.
8
Factorising by inspection
Now think about the x² term. When you multiply
out the brackets, you get two x² terms.
-3 multiplied by x² gives 3x²
x³ x² 4x - 6 (x 3)(x² 2x 2)
x multiplied by bx gives bx²
So 3x² bx² -1x²
therefore b must be 2.
9
Factorising by inspection
You can check by looking at the x term. When you
multiply out the brackets, you get two terms in x.
-3 multiplied by 2x gives -6x
x³ x² 4x - 6 (x 3)(x² 2x 2)
x multiplied by 2 gives 2x
-6x 2x -4x as it should be!
10
Factorising by inspection
Now you can solve the equation by applying the
quadratic formula to x² 2x 2 0.
x³ x² 4x - 6 (x 3)(x² 2x 2)
The solutions of the equation are x 3, x -1
i, x -1 i.
11
Factorising polynomials
Click here to see this example of factorising by
inspection again
Click here to see factorising using a table
Click here to see polynomial division
Click here to end the presentation
12
Factorising using a table
If you find factorising by inspection difficult,
you may find this method easier.
Some people like to multiply out brackets using a
table, like this
2x³
-6x²
-8x
3x²
-9x
-12
So (2x 3)(x² - 3x 4) 2x³ - 3x² - 17x - 12
The method you are going to see now is basically
the reverse of this process.
13
Factorising using a table
If you divide x³ - x² - 4x - 6 (cubic) by x 3
(linear), then the result must be quadratic.
Write the quadratic as ax² bx c.
14
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
The only x³ term appears here,
so this must be x³.
15
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
This means that a must be 1.
16
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
This means that a must be 1.
17
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
-6
The constant term, -6, must appear here
18
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
-6
so c must be 2
19
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
-6
so c must be 2
20
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
-3x²
-6
Two more spaces in the table can now be filled in
21
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
2x²
-3x²
-6
This space must contain an x² term
and to make a total of x², this must be 2x²
22
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
2x²
-3x²
-6
This shows that b must be 2
23
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
2x²
-3x²
-6
This shows that b must be 2
24
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
2x²
-3x²
-6
-6x
Now the last space in the table can be filled in
25
Factorising using a table
The result of multiplying out using this table
has to be x³ - x² - 4x - 6
x³
2x
2x²
-3x²
-6
-6x
and you can see that the term in x is -4x, as it
should be.
So x³ - x² - 4x - 6 (x 3)(x² 2x 2)
26
Factorising by inspection
Now you can solve the equation by applying the
quadratic formula to x²- 2x 2 0.
x³ x² 4x - 6 (x 3)(x² - 2x 2)
The solutions of the equation are x 3, x -1
i, x -1 i.
27
Factorising polynomials
Click here to see this example of factorising
using a table again
Click here to see factorising by inspection
Click here to see polynomial division
Click here to end the presentation
28
Algebraic long division
Divide x³ - x² - 4x - 6 by x - 3
x - 3 is the divisor
x³ - x² - 4x - 6 is the dividend
The quotient will be here.
29
Algebraic long division
First divide the first term of the dividend, x³,
by x (the first term of the divisor).
x²
This gives x². This will be the first term of the
quotient.
30
Algebraic long division
x²
Now multiply x² by x - 3
and subtract
31
Algebraic long division
x²
- 4x
Bring down the next term, -4x
32
Algebraic long division
Now divide 2x², the first term of 2x² - 4x, by x,
the first term of the divisor
x²
2x
- 4x
which gives 2x
33
Algebraic long division
x²
2x
- 4x
Multiply 2x by x - 3
2x² - 6x
2x
and subtract
34
Algebraic long division
x²
2x
- 6
- 4x
Bring down the next term, -6
2x² - 6x
2x
35
Algebraic long division
x²
2x
2
Divide 2x, the first term of 2x - 6, by x, the
first term of the divisor
- 4x
2x² - 6x
2x
- 6
which gives 2
36
Algebraic long division
x²
2x
2
- 4x
2x² - 6x
Multiply x - 3 by 2
2x
- 6
Subtracting gives 0 as there is no remainder.
2x - 6
0
37
Factorising by inspection
So x³ x² 4x - 6 (x 3)(x² - 2x 2)
Now you can solve the equation by applying the
quadratic formula to x²- 2x 2 0.
The solutions of the equation are x 3, x -1
i, x -1 i.
38
Factorising polynomials
Click here to see this example of polynomial
division again
Click here to see factorising by inspection
Click here to see factorising using a table
Click here to end the presentation