QUESTIONS: 2014; 2c 2013; 1c ALGEBRAIC LONG DIVISION x + n ax 3 + bx 2 + cx +d Used to find factors and remainders of functions for instance 2x 3 + 9x 2 + 8x + p This process is useful for finding factors when one value in the given function is unkown Read the question carefully to determine whether you need to give the factors or the roots Write out the equation with values ordered in descending powers of x Ensure each power of x is represented, if not write them in as 0x n If an equation is divided by a factor the remainder will be 0 Questions will provide one factor of the equation as a starting point If a complex number (a + bi) is a factor, its conjugate (a - bi) will also be a factor This process can be used to find remainders, however, the Remainder Theorem is a much quicker way to do this
PRACTICE QUESTION Step One If 2x 3 + 9x 2 + 8x + p has the root -3 + 2i find the value of p and the other two roots Step Two Determine the second root as the conjugate of the first root x = -3-2i Multiply the first two factors together Divide the original equation by the product of the first two factors Remainder must be zero so (x + 3-2i)(x + 3 + 2i) x 2 + 6x + 13 2x - 3 x 2 + 6x + 13 2x 3 + 9x 2 + 8x + p - (2x 3 +12x 2 +26x) 3x 2-18x + p - (3x 2 +18x +39) p - 39 p = 39 studytime.co.nz facebook.com/studytimenewzealand The result of the division gives the third factor So the third root is 2x - 3 x = 3/2
QUESTIONS: 2014; 1b, 1d, 2b 2013; 1d, 1e, 2b, 3d COMPLEX NUMBERS POLAR FORM z = rcis(θ) Used to represent the modulus and arguement of complex numbers for instance 4cis(180) The letter z is used to represent a complex number cos(θ) + i.sin(θ) is shortened to cis(θ) r represents the modulus of z, this can be written as I z l = r Θ represents the arguement of z, this can be written as arg(z) = Θ Θ should be expressed as an angle between -180 o and 180 o, if your answer is not between these values, add or subtract 360 o until it is Complex numbers can be represented visually using an Argand diagram r represents the distance from the origin Θ represents the angle anti-clockwise from the horizontal real axis To multiply numbers in polar form, multiply moduli and add arguements To divide numbers in polar form, divide moduli and minus arguements
PRACTICE QUESTION a = uv If u = 4cis(180) and v = 2cis(90) find and b = u v represent on an Argand diagram Step One Find the values of a and b Step Two Plot points on an Argand diagram Find uv multiply moduli and add arguements minus 360 o to get angle between 180 o and -180 o Find u v divide moduli and minus arguements a = 4cis(180). 2cis(90) a = 8cis(270) a = 8cis(-90) b = 4cis(180) 2cis(90) b = 2cis(90) Im(z) b Re(z) studytime.co.nz facebook.com/studytimenewzealand a
QUESTIONS: 2014; 1b, 1e, 2b, 2c, 2d, 2e, 3b, 3d 2013; 1b, 1c, 1d, 1e, 2e, 3a, 3b COMPLEX NUMBERS RECTANGULAR FORM z = x + iy Used to represent the real and imaginary parts of a complex number for instance 2 + 3i The letter z is used to represent a complex number Complex numbers have real and imaginary parts x is the real part of a complex number y is the coeficient of i, the imaginary part of a complex number The letter i is used to represent -1 The square root of a negative number does not exist so i is an imaginary number Complex numbers can be represented visually using an Argand diagram The value of x represents the distance along the horizontal or real axis The value of y represents the distance along the vertical or imaginary axis A line can be drawn from the origin to the x,y coordinate
PRACTICE QUESTION If a = u + v u = 1-2i and v = 2 + 3i find and b = uv Step One represent on an Argand diagram Step Two Find the values of a and b Plot points on an Argand diagram Find u+v Collect like terms a = (1-2i) + (2 + 3i) a = 3 + i Im(z) Find u.v b = (1-2i)(2 + 3i) a Multiply all terms Simplify b = 2 + 3i - 4i - 6i 2 b = 8 - i b Re(z) studytime.co.nz facebook.com/studytimenewzealand
QUESTIONS: 2014; 2c, 2e, 3d CONJUGATES 2013; 1a, 1c, 2d, 2e, 3b a + bi = a - bi a + b = a - b Used to rationalise complex numbers and surds for instance 4-2i 3 + i A fraction with an imaginery number or a surd in the denominator is an irrational number When a complex number or a surd is multiplied by its conjugate, the result is a rational number The notation for the conjugate of a number, z, is z Remember to express answers in the form asked for in the question Remember that ( A) 2 = A and i 2 = -1 If a complex number is the root of a quadratic, its conjugate must also be a root Remember to only change the sign in front of the complex number or surd when writing the conjugate of a number The method of eliminating an imaginery number or surd by multiplying by its conjugate is the same as the difference of two squares
PRACTICE QUESTION Write 4-2i 3 + i in the form a + bi Step One Step Two Write out the conjugate 3 + i = 3 - i studytime.co.nz facebook.com/studytimenewzealand Multiply the fraction by one Write 1 as an equivalent fraction using the conjugate Expand the brackets Simplify Write in the form a + bi 4-2i. 1 3 + i 4-2i. 3 - i 3 + i 3 - i (4-2i)(3 - i) (3 + i)(3 - i) 12-4i - 6i + 2i 2 9-3i + 3i - i 2 10-10i 10 1 - i
QUESTIONS: 2014; 1b, 1d, 2b, 2e 2013; 1d, 1e CONVERSION POLAR AND RECTANGULAR FORMS rcis(θ) = a + bi Used to convert complex numbers into a given form for instance express 5cis(30) in rectangular form a + bi is rectangular or cartesian form rcis(θ) is polar form or modulus arguement form Use trigonometry to convert between forms Always sketch a diagram r b Θ a Rectangular to Polar: Θ = tan -1 b/a r = (a 2 + b 2 ) Polar to Rectangular: a = rcos(θ) b = rsin(θ) Conversions can be done on a graphics calculator
PRACTICE QUESTION A. If B. If z = 5cis(30) express z in rectangular form z = 4 + 2i express z in polar form Step One Step Two A. Sketch a diagram 5 30 o a b A. Find a Find b a = 5. cos(30) a = 4.33 (3sf) b = 5. sin(30) b = 2.50 (3sf) z = 4.33 + 2.50i B. r Θ 4 2 B. Find r Find Θ r = (2 2 + 4 2 ) r = 4.47 (3sf) Θ = tan -1 (2/4) Θ = 26.6 o (3sf) studytime.co.nz facebook.com/studytimenewzealand z = 4.47cis(26.6)
QUESTIONS: 2014; 1b, 1d, 2b DE MOIVRE S THEOREM 2013; 3d [rcis(θ)] n = r n cis(nθ) Used to find the power of an equation of the form: rcis(θ) for instance [3cis(120)] 5 De Moivre s Theorem is used to find powers and roots of complex numbers This method can only be used for complex numbers in polar form If a complex number is given in rectangular form it may easier to convert it into polar form to find its power or root To find the n root of a number use: r 1/n cis(θ/n) Θ should be expressed as an angle between -180 o and 180 o, if your answer is not between these values, add or subtract 360 o until it is If you are unsure refer to De Moivre s theorem on the formula sheet r represents the modulus Θ represents the arguement
PRACTICE QUESTION If u = 3cis(120) find u 5 Step One Write out the equation Step Two So we can sub this into [rcis(θ)] n = r n cis(nθ) u 5 = [3cis(120)] 5 Simplify minus 360 o until angle is between 180 o and -180 o u 5 = 3 5 cis(120. 5) u 5 = 243cis(600) u 5 = 243cis(600-360 - 360) studytime.co.nz facebook.com/studytimenewzealand u 5 = 243cis(-120)
QUESTIONS: 2014; 1e 2013; 2e LOCI z = x + iy Geometric representation of complex numbers for instance z - (4 - i) = 10 A locus is a path which can represent a straight line, circle, ellipse, hyperbola or parabola, these are referred to as conic sections The aim of this process is to describe the path of the locus in terms of how it could be graphed In your answer state the shape of the path as well as key points such as the centre and radius for a circle Use z = x + iy (rectangular form) to split z into real and imaginary parts Always separate and collect real and imaginary terms x + iy represents the modulus of the locus, which can be written as (x 2 + y 2 ) by converting to polar form A circle with a centre (a,b) and radius r has the equation (x - a) 2 + (y -b) 2 = r 2 For more information on paths and their equations refer to the conic sections internal
PRACTICE QUESTION Fully decribe the locus of z if z - (4 - i) = 10 Step One Step Two Write z as x + iy (x + iy) - (4 - i) = 10 Collect real and imaginary terms Re-write in polar form Square both sides and get The equation is of the form Therefore the locus can be described as (x - 4) + (iy + i) = 10 (x - 4) 2 + (y + 1) 2 = 10 (x - 4) 2 + (y + 1) 2 = 10 2 (x - a) 2 + (y -b) 2 = r 2 A circle with a centre at (4,-1) and a radius of 10 studytime.co.nz facebook.com/studytimenewzealand
QUESTIONS: 2014; 3c QUADRATIC FORMULA 2013; 3e = b 2-4ac Used to find values in an equation that satisfy a given condition for instance x 2 + kx +k When > 0 the equation has two distinct real roots When = 0 the equation has one distinct real root When < 0 the equation no real roots A root is where the function intersects the x-axis is the discriminant of the quadratic equation: x = -b + (b 2 - - 4ac) 4ac This applies to quadratics of the form ax 2 + bx + c If you are unsure refer to quadratics on the formula sheet The solution may contain a distinct value or a range of values that satisfy the given condition Conditions relate to the number of roots a function is desired to have, or how many times the graph intersects the x-axis
PRACTICE QUESTION If x 2 + 5x - k find the values of k for which the function does not intersect the x axis Step One Step Two Determine the condition If <0 then b 2-4ac < 0 < 0 Substitute the values from the question 5 2 - (4. 1. -k) < 0 25 + 4k < 0 Solve the equation 4k < -25 k < -25 4 studytime.co.nz facebook.com/studytimenewzealand
QUESTIONS: 2014; 1c, 2a, 3a SURDS 2013; 1a, 3a, 3c, 3e a b Used to express the irrational root of an integer for instance 72 + 50 A surd is an irrational number It is easiest to simplify surds before manipulating them Addition or subtraction can only take place if the numbers under the surd signs are the same: a+ a = 2 a Multiplication does not require the numbers under the surd signs to be the same: a. b = ab Express answers with surds in their simplest form Look for factors that are perfect squares to remove from the surd: 8 = 2.2 2 = 2 2 To simplify numbers with exponents, halve the power and leave the remainder under the surd sign: x 5 = x 2 x If you get stuck trying to find factors, go through the perfect squares until you find one that works, ie. 4, 9, 16, 25, 36, 49 etc.
PRACTICE QUESTION Find the sum, difference and product of 72 and 50 Step One Find the common factors 72 = 2. 36 50 = 2. 25 Step Two Simplify the surds by removing the perfect squares Find the sum 72 = (2. 6 2 ) 50 = (2. 5 2 ) 72 = 6 2 50 = 5 2 6 2 + 5 2 studytime.co.nz facebook.com/studytimenewzealand Find the difference Find the product 11 2 6 2-5 2 2 6 2. 5 2 30. 2 = 60