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Polynomials & Cubic Polynomials & Cubic FactorsFactors
© Ciarán Duffy
Lucan Community Lucan Community CollegeCollege
Leaving Certificate Leaving Certificate MathematicsMathematicsHigher LevelHigher Level
Mr DuffyMr Duffy
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
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
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.
