10: polynomials © christine crisp “teach a level maths” vol. 1: as core modules

10: Polynomials10: Polynomials

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Polynomials

Module C1

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Polynomials

Polynomials only contain terms of the type ,

where n is a positive integer

nax

The following are examples of Polynomial Functions:

Polynomial

Functions

10132 23 xxx

11324 xxx

32 xx A quadratic polynomial

A cubic polynomial

A quartic polynomial

Polynomials Expanding Cubic

Functions

e.g. 1

))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged

))(( 1 x10x2 x52x

Polynomials

e.g. 1

))()(( 125 xxxWe 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

))(( 1 x10x2 x52x

))(( 11032 xxx

Expanding Cubic Functions

Polynomials

))(( 1 xxx 1032

e.g. 1

))()(( 125 xxxWe 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

xxx 103 23

))(( 1 x10x2 x52x

Expanding Cubic Functions

Polynomials

))((

e.g. 1

))()(( 125 xxxWe 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

xxx 103 23

))(( 1 x10x2 x52x

xxx 1032 1

10132 23 xxx

1032 xx

Expanding Cubic Functions

Polynomials

Expand the brackets in the following: )2)(1)(3( xxx

)2)(33( 2 xxxx

))(( 2322 xxx

642

322

23

xx

xxx

673 xx

Answer:

673 xx

Solution:

Exercise

)2)(1)(3( xxx

Polynomials Factorising Simple

Cubics

e.g. Factorise fully the following: xxxxf 54)( 23

Common factor:

Solution:

( Others are best done using the Factor Theorem which is covered later ).

Some cubic functions which contain a common factor can be factorised by inspection.

)54()( 2 xxxxf

Trinomial factors:

)1)(5()( xxxxf

We must now factorise the quadratic.

Polynomials

Factorise fully the following cubic:

xxx 82 23

)82( 2 xxx

)4)(2( xxx

Solution:

Exercise

xxx 82 23

Polynomials

Polynomials

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Polynomials

Polynomials only contain terms of the type ,

where n is a positive integer

nax

The following are examples of Polynomial Functions:

Polynomial

Functions

10132 23 xxx

11324 xxx

32 xx A quadratic polynomial

A cubic polynomial

A quartic polynomial

Polynomials

Expanding Cubic Functions

e.g. 1

))()(( 125 xxxWe 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

xxx 103 23

))(( 1 x10x2 x52x

)1)(103( 2 xxx

10132 23 xxx

1032 xx

Polynomials

Factorising Simple Cubics

e.g. Factorise fully the following: xxxxf 54)( 23

Common factor:

Solution:

( Others are best done using the Factor Theorem which is covered later ).

Some cubic functions which contain a common factor can be factorised by inspection.

)54()( 2 xxxxf

Trinomial factors:

)1)(5()( xxxxf

We must now factorise the quadratic.