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“Teach A Level Maths”
Vol. 1: AS Core Modules
10: Polynomials
© Christine Crisp
Polynomials
Module C1
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Polynomials
Polynomial Functions
The following are examples of Polynomial Functions:
x  x3
2
x 3  2 x 2  13 x  10
x  x  13 x  1
4
2
A quadratic polynomial
A cubic polynomial
A quartic polynomial
Polynomials only contain terms of the type ax n,
where n is a positive integer
Polynomials
Add 2x3 + 3x2 – 4 and x2 – x -2.
Polynomials
Find the x and x2 terms in the product of
(2x + 3) (3x2 -2x +4).
Polynomials
If (Ax + B) (2x – 9) = (6x2 - 19x - 36), find A and B.
Polynomials
Expanding Cubic Functions
e.g. 1
( x  5)( x  2)( x  1)
 (x 2  2 x  5 x  10)( x  1)
We multiply 2 of the
parts together first,
leaving the third
unchanged
Polynomials
Expanding Cubic Functions
e.g. 1
( x  5)( x  2)( x  1)
 (x 2  2 x  5 x  10)( x  1)
 ( x 2  3 x  10)( x  1)
We multiply 2 of the
parts together first,
leaving the third
unchanged
Now multiply each of
the 3 terms in the 1st
pair of brackets by
each of the 2 terms
in the 2nd
Polynomials
Expanding Cubic Functions
e.g. 1
( x  5)( x  2)( x  1)
 (x 2  2 x  5 x  10)( x  1)
 ( x 2  3 x  10)( x  1)
 x 3  3 x 2  10 x
We multiply 2 of the
parts together first,
leaving the third
unchanged
Now multiply each of
the 3 terms in the 1st
pair of brackets by
each of the 2 terms
in the 2nd
Polynomials
Expanding Cubic Functions
e.g. 1
( x  5)( x  2)( x  1)
 (x 2  2 x  5 x  10)( x  1)
 ( x 2  3 x  10)( x  1 )
 x 3  3 x 2  10 x
 x 2  3 x  10
 x 3  2 x 2  13 x  10
We multiply 2 of the
parts together first,
leaving the third
unchanged
Now multiply each of
the 3 terms in the 1st
pair of brackets by
each of the 2 terms
in the 2nd
Polynomials
Exercise
Expand the brackets in the following:
( x  3)( x  1)( x  2)
Solution:
( x  3)( x  1)( x  2)
 ( x 2  x  3 x  3)( x  2)
 ( x 2  2 x  3)( x  2)
 x3  2x2  3x
 2x2  4x  6
 x3
 7x  6
Answer:
x  7x  6
3
Polynomials
Factorising Simple Cubics
Some cubic functions which contain a common
factor can be factorised by inspection.
( Others are best done using the Factor Theorem
which is covered later ).
e.g. Factorise fully the following:
f ( x)  x 3  4 x 2  5 x
Solution:
Common factor:
2
f
(
x
)

x
(
x
 4 x  5)

We must now factorise the quadratic.
Trinomial factors: 
f ( x)  x( x  5)( x  1)
Polynomials
Exercise
Factorise fully the following cubic:
x  2x  8x
3
Solution:
2
x 3  2x 2  8x
2
 x ( x  2 x  8)
 x( x  2)( x  4)
Polynomials
Summary
The degree of a polynomial is the highest x power
in the expression.
Add or subtract polynomials by column addition or
subtraction, or by collecting like terms.
Multiply polynomials using any method that helps
you to remember to multiply every term in one
expression by every term in the other.
Solve identities by equating co-efficients.
Polynomials
Practice Problems
Exercise 9 – choose questions to work from in
class and at home, but this is a key topic to help
you improve your algebra.
If this topic is an area of weakness for you,
complete all questions.
Mark them yourselves, and bring to me any
problems that are wrong but you don’t know why.