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Higher'Degree'Polynomials' '''''''''''' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''Page'1'of'16'

! ∙ #$ =&'(

) ∙ #$ = 0

! ∙ #+ = −#Φ. #/

) ∙ #+ = 0(1 + 0('(#Φ3 #/

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MATHEMATICAL'METHODS'UNIT!1'CHAPTER'5'–'HIGHER'DEGREE'POLYNOMIALS'Key'knowledge'

•! The'key'features'and'properties'of'cubic'polynomials'functions'and'their'graphs' '•! The' effect' of' transformations' of' the' plane,' dilation,' reflection' in' axes,' translation' and' simple'

combinations'of'these'transformations,'on'the'graphs'of'cubic'polynomials' '•! The'definition'of'a'function,'the'concepts'of'domain,'coNdomain'and'range,'notation'for'specification'of'

the'domain'(including'the'concept'of'maximal,'natural'or'implied'domain),'coNdomain'and'range'and'rule'of'a'function' '

Key'skills'

•! use'algebraic,'graphical'and'numerical'approaches,'including'the'factor'theorem'to'determine'and'verify'solutions'to'equations'over'a'specified'interval'

•! sketch'by'hand'graphs'of'cubic'polynomial'functions,'in'factored'form,'including'cases'where'an'4Naxis'intercept'is'a'touch'point'or'a'stationary'point'of'inflection' '

CHAPTER 5 – SET QUESTIONS EXERCISE!5.2:'QUARTIC'POLYNOMIALS'

2,'4,'6,'7a,'8d,'11,'12ace,'13bcd,'15,'17,'18''

EXERCISE!5.3:'FAMILIES'OF'POLYNOMIALS'

1,'4,'5,'8,'9a,'10a,'11,'12,'16,'17,'19,20'

EXERCISE!5.4:'NUMERICAL'APPROXIMATIONS'TO'ROOTS'OF'POLYNOMIAL'EQUATIONS'

1,'4,'6,'8ab,'11,'13ab,'14ac,'15,'17,'18'

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Table&of&Contents'

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CHAPTER!5!–!HIGHER!DEGREE!POLYNOMIALS! 1'KEY'KNOWLEDGE' 1'KEY'SKILLS' 1'

5.2!QUARTIC!POLYNOMIALS! 3'GRAPHS!OF!QUARTIC!POLYNOMIALS!OF!THE!FORM!Y6 = A(X6:6H)4 + C! 3'

Example'1'(Q8a)' 3'QUARTIC!POLYNOMIALS!WHICH!CAN!BE!EXPRESSED!AS!THE!PRODUCT!OF!LINEAR!FACTORS! 4'

Example'2' 5'EQUATIONS'AND'INEQUATIONS' 5'

Example'3'(Q5)' 6'

5.3!FAMILIES!OF!POLYNOMIALS! 7'GRAPHS!OF!Y6 = 6XN,!WHERE!N ∈ N!AND!N!IS!ODD! 7'

Example'4'(Q2)' 7'GRAPHS!OF!Y6 = 6XN,!WHERE!N!∈ N!AND!N!IS!EVEN ! 8'

Example'5'(Q3)' 8'FAMILIES!OF!POLYNOMIALS!WHICH!CAN!BE!EXPRESSED!AS!THE!PRODUCT!OF!LINEAR!FACTORS! 9'

EFFECT'OF'MULTIPLICITY'OF'ZEROS'AND'LINEAR'FACTORS' 9'Example'6'(Q6)' 9'

OTHER!FAMILIES!OF!POLYNOMIALS! 10'Example'7'(Q7)' 10'

5.4!NUMERICAL!APPROXIMATIONS!TO!ROOTS!OF!POLYNOMIAL!EQUATIONS! 11'EXISTENCE'OF'ROOTS' 11'THE'METHOD'OF'BISECTION' 11'

Example'8'(Q2)' 12'USING!THE!INTERSECTIONS!OF!TWO!GRAPHS!TO!ESTIMATE!SOLUTIONS!TO!EQUATIONS! 13'

Example'9'(Q'12)' 14'ESTIMATING!COORDINATES!OF!TURNING!POINTS! 15'

AN'ALTERNATIVE'APPROACH' 15'Example'10'(Q14ab)' 16'

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5.2&Quartic&Polynomials&A'quartic'polynomial'is'a'polynomial'of'degree'4'and'is'of'the'form:'

A(4) = B44

6

+ C43

6

+ E42

6

+ #4 + G,'where'B ≠ 06BI#6B6, C, E, #, G ∈ K.'

Graphs!of!quartic!polynomials!of!the!form!L6 = M(N6 − 6O)P + Q!!

The' simplest' quartic' polynomial' graph' has' the' equation' R = 44.' As' both' negative' and' positive'numbers'raised'to'an'even'power,'in'this'case'4,'will'be'positive,'the'longN'term'behaviour'of'the'graph'of'R = 44'must'be'that'as'4 → −∞'or'as'46 → ∞,'then'R → ∞.''

The'graph'of'R6 = 644'is'like that'of'the'parabola'R6 = 642.'Both'graphs'are'concave'up'with'a'minimum'turning'point'at'(0, 0)'and'both'contain'the'points'(−1, 1)'and'(1, 1).'However,'for'the'intervals'where'x"<'−1'and'x">'1,'the'graph'of'R6 = 644'lies'above'the'parabola.'This'is'because'44 > 42'for'these'intervals.'Likewise,'the'graph'of'R6 = 644'lies'below'that'of'the'parabola'for'the'intervals'−1 < 4 < 0'and'0 < 64 <1,'since'44 < 642'for'these'intervals.''

Under'a'dilation'of'a"units,'a'horizontal'translation'of'h"units'and'a'vertical'translation'of'k"units,'the'graph'of'R6 = 644'is'transformed'to'that'of''R6 = 6B(46 − 6ℎ)4

6

+ 6Y.''

Example'1'(Q8a)'

Sketch'the'following'graph,'identifying'the'coordinates'of'the'turning'point'and'any'point'of'intersection'with'the'coordinate'axes.'R = (4 − 1)Z −16'

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On the CAS On'a'blank'graphs'page'c12''Press'.6(relation)'Type:'R = (4 − 1)Z − 16then'press'· Note:'We'may'need'to'reNsize'the'graph.'To'do'this'press'·41'or'another'suitable'option.'

On'the'displayed'graph,'we'wish'to'find:''The'xNintercepts:'press'b61,'then'use'the'arrow'keys'to'select'a'point'left'of'the'first'xNintercept'press'·'move'to'the'right'of'the'xNintercept'and'press'·.'Repeat'for'the'other'xNintercepts.'The'turning'point:'press'b62'(minimum,'in'this'case),'then'use'the'arrow'keys'to'select'a'point'left'of'the't.p.'·'move'to'the'right'of'the't.p.'and'press'·.'The'yNintercept:'On'a'calculator'page,'solve(R = (4 − 1)Z − 16, R)|4 = 0'gives'R = −15'

Quartic!polynomials!which!can!be!expressed!as!the!product!of!linear!factors!!Not'all'quartic'polynomials'have'linear'factors.'However,'the'graphs'of'those'which'can'be'expressed'as'the'product'of'linear'factors'can'be'readily'sketched'by'analysing'these'factors.''

'Given'the'longNterm'behaviour'of'a'quartic'polynomial'whereby'R6 → 6∞6as646 → 6±∞'for'a'positive'coefficient'of'the'term'in'44,'the'sign'diagrams'and'accompanying'shape'of'the'graphs'must'be'of'the'form'shown'in'the'diagrams.'

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'For'a'negative'coefficient'of'44,'R → −∞6as64 → ±∞,'so'the'sign'diagrams'and'graphs'are'inverted.'


'Example'2'

Sketch'the'following'graphs'by'determining'the'x'and'y'intercepts'(don’t'attempt'calculate'the'turning'points).'

a)'R = (4 − 2)a(4 − 1)'

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b)6R = (4 − 3)(4 − 1)(4 + 1)b''

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Check on the on the CAS On'graphs'page'c12''Press'.6(relation)'Type:'R = 4 − 2 a 4 − 1 6then'press'· Repeat'for''R = (4 − 3)(4 − 1)(4 + 1)b''Note:'We'may'need'to'reNsize'the'graph.'To'do'this'press'·41'or'another'suitable'option.'

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equations'and'inequations''

If' a' quartic' polynomial' A(4)' can' be' rewritten' as' the' product' of' linear' factors' i.e.'64 − B 4 − C 4 − E 4 − # .'Its'graph'R = A(4)'can'then'be'readily'sketched'from'this'form'and'with'the'aid'of'a'sign'diagram,'or'a'graph,'an'inequation'such'as'P(x)'≤'0'can'be'solved.''

1.! Find'an'4 = B'for'A B = 0,'use'this'(4 − B)'as'a'linear'factor'and'long'division'to'find'a'cubic'quotient''

2.! Further'use'of'the'factor'theorem'may'enable'the'finding'of'a'quadratic'quotient'leading'to'further'factorization'to'find'linear'factors.'

OR''

3.! Find'an'4 = B'and'4 = C'for'A(B) = 0'and'A(C) = 0'and'then'use'the'product'of'(4 − B)(4 − C)'and'long'division'to'find'a'quadratic'quotient.''

4.! Further'factorization'to'find'linear'factors.'

Example'3'(Q5)'

Factorise'A 4 = 44 + 543 − 642 − 324 + 32,'hence'solve'the'inequation'44 + 543 − 642 − 324 +32 > 0.'

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On the CAS On'a'blank'calculator'page'c11'''Type:'solve 4Z + 54a − 64b − 324 + 32 > 0, 4 6then'press'·

The'answer'is'given'as'4 ≠ −46BI#64 < 16hi64 > 2''Or"in"set"notation:"The'answers'is:'

4: 4 < −4 6∪ 6 4: −4 < 4 < 1 ∪6 4: 4 > 2 '

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