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Algebra

TABLE OF CONTENTSQUESTION 1..................................................................................................................................1QUESTION 2..................................................................................................................................2QUESTION 3..................................................................................................................................5QUESTION 4..................................................................................................................................7

QUESTION 1x² – 2x -1 = 0Range of roots: 2.3 <α < 2.5i)Show the working to arrive at an iterative formulaEquation given = x² – 2x -1 = 0x² = 2x +1On dividing both the sides by x we getx = 2 + (1/x)Thus iterative formula can be written as :x(n+1) = 2 + [ 1/x(n) ]ii)Include a table to clearly demonstrate the iterative process.(You are only required to demonstrate the values up to the point whereby your answer iscorrect to 4 d.p.)x(n+1) = 2 + [ 1/x(n) ]Let the initial value x0 = 2.4NXnXn +10X02.41X1 = 2 + (1 / 2.4)2.41662X2 = 2 + (1 / 2.4166)2.41383X3 = 2 + (1 / 2.4138)2.41424X4 = 2 + (1 / 2.4138)2.41421

iii)Explain concisely why you have chosen to stop the iterative process where you haveAfter fourth iteration the process have been chosen to stop because both third and fourth iterationhave same value up to three decimal points.iv)use your formula to find the root correct to 4 decimal places.(You should choose your starting value from the given information)x = 2 + (1/x)Let the initial value is x0 = 2.4From the above table it can be observed that upto 4 iterations are possible thus root isgiven by 2.4142QUESTION 2a)Factorise fully the following cubic;x3+ 4x2+ x – 6(Ensure you demonstrate relevant working regarding how you obtain your fist rootand any obtained quadratic)Answer:Detailed solution :Step1: Equation at the end of step1= (((x3) + 22x2) + x) - 6Step2: Checking for a perfect cube :2.1: x3+4x2+x-6Referring this, it can be stated that itis not a perfect cube2.2Factoring:x3+4x2+x-6Expressionsaredividedintotwogroups:2

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