Additional_Mathematics_Notes

Additional Mathematics Notes Functions and Graphs 2𝑥#+2 𝑎>0 Turning point = Minimum point −2𝑥#+2 𝑎<0 Turning point = Maximum point Line of symmetry always passes through the turning point (𝑥-coordinate of turning point/mid-value of 𝑥-intercepts) y-intercept, x=0 𝑦=𝑎𝑥#+𝑏𝑥+𝑐 𝑦=𝑎(0)#+𝑏(0)+𝑐 x-intercept, y=0 0=𝑎𝑥#−𝑏𝑥+𝑐 Completing the Square 𝑎𝑥#+𝑏𝑥+𝑐 𝑎(𝑥#+𝑏𝑎𝑥+𝑐𝑎) 𝑎0(𝑥+𝑏2𝑎)#−1𝑏2𝑎2#+𝑐𝑎3 𝑎(𝑥+ℎ)#+𝑘 𝑎(𝑥−ℎ)#+𝑘 Coefficient of 𝑥# must be +1 𝑎𝑥#−𝑏𝑥+𝑐 𝑎(𝑥#−𝑏𝑎𝑥+𝑐𝑎) 𝑎0(𝑥−𝑏2𝑎)#−1𝑏2𝑎2#+𝑐𝑎3 Conditions for Curve to Lie Completely Above or Below x-axis Above Coefficient of 𝑥#>0 Minimum value is positive 𝑏#−4𝑎𝑐<0 ∴Since the coefficient of 𝑥#=1>0 and the minimum value of 𝑦=4, which is positive, the curve lies completely above the x-axis. Below Coefficient of 𝑥#<0 Minimum value is negative ∴Since the coefficient of 𝑥#=−1<0 and the maximum value of 𝑦=−4, which is negative, the curve lies completely below the x-axis. Quadratic Formula (Given) 𝑎𝑥#+𝑏𝑥+𝑐 𝑥=−𝑏±√𝑏#−4𝑎𝑐2𝑎 Nature of Roots of Quadratic Equations 𝑏#−4𝑎𝑐≥0 Real roots 𝑏#−4𝑎𝑐>0 2 real and distinct roots 𝑏#−4𝑎𝑐<0 Non-real roots 𝑏#−4𝑎𝑐=0 2 real and equal roots Intersection Between Line and Curve 𝑏#−4𝑎𝑐>0 Line cuts curve at 2 distinct points 𝑏#−4𝑎𝑐=0 Line cuts curve at 1 point (Line is tangent to curve) 𝑏#−4𝑎𝑐<0 Line does not intersect curve Finding Range of Values E.g. (𝑝−6)(𝑝+2)>0 Draw number line and curve Laws of Surds √𝑎𝑏 √𝑎×√𝑏 ?𝑎𝑏 √𝑎√𝑏 √𝑎×√𝑎, (√𝑎)# 𝑎 𝑝√𝑎+𝑞√𝑎 (𝑝+𝑞)√𝑎 𝑝√𝑎−𝑞√𝑎 (𝑝−𝑞)√𝑎 𝑝#+(𝑞√𝑎)# 𝑝#+𝑞#𝑎 1√2+3 1√2+3×√2−3√2−3 √2−32−9 −√2−37 Equality of Surds 𝑎+𝑏√𝑛 5+4√𝑛 𝑎=5 𝑏=4

Polynomials Degree of 𝑃(𝑥)×𝑄(𝑥) Degree of 𝑃(𝑥) + Degree of 𝑄(𝑥) Division Algorithm: 𝑃(𝑥)=𝐷(𝑥)×𝑄(𝑥)+𝑅(𝑥) 𝑃(𝑥) = Dividend 𝐷(𝑥) = Divisor 𝑄(𝑥) = Quotient 𝑅(𝑥) = Remainder Remainder and Factor Theorem 𝑓(𝑥)÷(𝑎𝑥+𝑏) 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟=𝑓(−𝑏𝑎) Factor Theorem, Remainder=0 Cubic Expressions and Equations 𝑎𝑥R+𝑏𝑥#+𝑐𝑥+𝑑 (𝑎𝑥+𝑏)(ℎ𝑥+𝑘)(𝑝𝑥+𝑞) (𝑎𝑥+𝑏)(𝑝𝑥#+𝑞𝑥+𝑟) Step 1 Factorise 𝑓(𝑥) using Factor Theorem [Mode-3-4] Step 2 Use Synthetic Division/ Long Division/ Comparing Coefficients to factorise 𝑓(𝑥) completely Synthetic Division 𝑓(𝑥)=2𝑥R−11𝑥#−7𝑥+6 -1 2 -11 -7 6 -2 13 -6 2 -13 6 0 (𝑥+1) 𝑓(𝑥)=(𝑥+1)(2𝑥#−13𝑥+6) Cubic Identities Sum of cubes 𝑎R+𝑏R=(𝑎+𝑏)(𝑎#−𝑎𝑏+𝑏#) Difference of cubes 𝑎R−𝑏R=(𝑎−𝑏)(𝑎#+𝑎𝑏+𝑏#) Partial Fractions Step 1: Fraction must be a proper fraction (Degree of numerator < Degree of denominator) Step 2: Factorise denominator Step 3: Express proper algebraic fraction in partial fractions Case Denominator Proper Fraction Partial Fraction 1 Distinct linear factors 𝑝𝑥+𝑞(𝑎𝑥+𝑏)(𝑐𝑥+𝑑) 𝐴𝑎𝑥+𝑏+𝐵𝑐𝑥+𝑑 2 Repeated linear factors 𝑝𝑥+𝑞(𝑎𝑥+𝑏)# 𝐴𝑎𝑥+𝑏+𝐵(𝑎𝑥+𝑏)# 3 Quadratic factor (cannot be factorised) 𝑝𝑥#+𝑞𝑥+𝑟(𝑎𝑥+𝑏)(𝑥#+𝑐#) 𝐴𝑎𝑥+𝑏+𝐵𝑥+𝐶𝑥#+𝑐# Step 4: Solve for unknown constant Binomial Theorem (Given) Binomial expansion of (𝑎+𝑏)V W𝑛0X𝑎V+W𝑛1X𝑎VYZ𝑏+W𝑛2X𝑎VY#𝑏#+⋯+W𝑛𝑟X𝑎VY\𝑏\+W𝑛𝑛−1X𝑎𝑏VYZ+(𝑛𝑛)𝑏V General Term 𝑇\^Z=W𝑛𝑟X𝑎VY\𝑏\ Law of Indices 𝑎_=1 𝑎`×𝑎V=𝑎`^V 𝑎YV=1𝑎V 𝑎`÷𝑎V=𝑎`YV 𝑎`V=√𝑎`a (𝑎`)V=𝑎`V (𝑎𝑏)V=𝑎V𝑏V (𝑎𝑏)V=𝑎V×𝑏