ZNOTES.ORGUPDATED TO 2023-25 SYLLABUSCAIE AS LEVELFURTHER MATHSSUMMARIZED NOTES ON THE FURTHER PURE 1 SYLLABUSCAIE AS LEVEL FURTHER MATHS=(a8+ay-By)-2(a+8+y)+31.Roots of Polynomial.-2-2(-3)+3=7Equations1.(a-1)(B-1)(y-1)=aBy-a8-ay+a-8y+8+y-1Quadratic Equations(ax2+bx+c =0)·∑a=a+B=-=aBy-(a8+ay+By)+(a+8+7)-1.-5-(-2)+(-3)+1=-7Cubic Equations (ax3+bx2+cx+d =0)Thus,equation is:x3-6x2+7x+7=01.3.SubstitutionFor Finding sums of roots to a specific degree of powerQuartic Equations (ax4+bx3+cx2+dx+e =0)(SP20-P01}Question 4:The Cubic Equation 23-22-z-5=0 has roots a,B and1.Show thata3+83+3 =19Recurrence Notation2.Find the value of a+B+·∑a2=(∑a)2-2∑y is also known as S2Solution:is also known as 1.It's always equal to theWe can use Recurrence Notation since:negative of the coefficient of the linear term divided bythe coefficient of the constant termFind*S**and **S2.*From the polynomial:1.2.Algebraic CombinationsS1=1Finding an equation through algebraic manipulationUsing S2=(∑a)2-2∑aBEx 1.3 Question 2b:x3+3x2-2x+5 =0 has roots a,B,YS2=(1)2-2(-1)=3Find equation with roots(a:-1),(B-1),(-1)From our polynomial,we know that:Solution:Using coefficients:S3-S2-S1-15=01.a+B+Y=-3Substitute in values and find SaS3-(3)-(1)-15=0Equation to find:→S3=15+3+11.(a-1)+(B-1)+(y-1)=→S3=19(Ans for a)=a+B+Y-3In order to find Sa,we will need to use a substitution,we willbe letting=z2→z=V万.-3-3=-6Rearrange the terms:1.(a-1)(B-1)+(a-1)(y-1)+(3-1)(y-1)Square both sides:=a8-a-8+1+ay-a-y+1+8Y-8-Y+1WWW.ZNOTES.ORGCAIE AS LEVEL FURTHER MATHSRearrange the terms back and substitute in z =y:2.5.Sign TablesUsed to visualize graph as it shows in which quadrant the→y3-3y2-9y-25=0graph liesEnter values of which result in different parts of theS2 in the new equation is Sain the old equationfraction equaling zeroLeave columns between each value of a and place signsS2=(3)2-2(-9)=27(Ans for b)to indicate whether value +ve or -ve in each cell2.Rational Functions3x2+3x+6y=(x+3)(x-2)2.1.Partial Fractions-32To split an improper fraction into partials,use PolynomialDivision and make sure the degree of the numerator is0+higher than the denominator.X-20If the degree of the numerator is less than thedenominator,use partial fraction method and equate+infinityinfinitycoefficient.2.6.Range of Function2.2.Vertical Asymptote·Discriminant:Making denominator 0 resulting in oo·b-4ac=0:Tangent◆Example:62-4ac 0 Lines do not meet(are not in range).B2-4ac >0:Lines do meet (are in range)1We can use the discriminant to show the Range of thefunction.Most of the time we use b2-4ac 0 to showwhat values of y that does not exist to then show whatThus,vertical asymptotes at:2 =-1 and 3values of y exists.2.3.Horizontal Asymptote2.7.Curve SketchingBy dividing the top and bottom of a fraction by z,we canWhen you sketch the curve,include the following:see what value y tends to when z becomes very large·-intercept◆Example:3x-2.Stationary points(maxima,minima,inflections)y=Vertical asymptote(s)Horizontal or oblique asymptote(s)Divide numerator and denominator byy=.{SP20-P01}Question 7:The Curve C has equationy=x2-2x+1a)State the equations of the asymptotes of CThus,horizontal asymptote at:y 3c)Find the coordinates of any stationary points of Cd)Sketch C,stating the coordinates of any intersections of C2.4.Oblique Asymptoteswith the coordinate axes and the asymptotesSolution:Occurs only with improper fractionIn order to find all the asymptotes,we will have to split the◆Example:improper fraction as much as possible:23Apply polynomial division to our function:y=2x-1+x-1x+2y=When x becomes very large,y 2x-12x2-3x-2(x-1)2Thus,oblique asymptote at:y 2x-1As x oo,y =2 which is our horizontal asymptoteWWW.ZNOTES.ORG