A Maths Tuition

What do we mean by the roots of a quadratic equation? Example 1 We say that 2 and 1 are roots of the equation. The equation has 2 real and distinct roots In graphical form, the roots of the equation are represented as the intersection of the curve with the x axis. Example 2 In this case there is only 1 solution. We say that the equation has only 2 real and equal roots Graphically it can be represented as Notice that there is only 1 intersection point between the curve and the x axis Example 3 Lastly we are going to look at an example where the curve does not intersect the x axis Consider the equation: The equation cannot be factorized. It cannot be solved using the quadratic formula. Graphically it is represented as It turns out that we can predict the number of roots that an equation has by looking at the discriminant Recall the quadratic formula

Question 1 The coefficient of the term in a cubic polynomial f(x) is -1. The roots of f(x) are 1, 2 and k. f(x) when divided by x - 3 gives a remainder of 8. Find the value of k First we need to understand what is meant by roots of an equation

Introduction How do we find the remainder when we divide by ? One way is to use long division From the long division, we get a remainder of 13 Note that: is known as the divisor is known as the quotient is known as the dividend In most cases we are only interested in the remainder, there is an easier way of obtaining the remainder without using long division The easier way is to use the Remainder Theorem Once again we want to find the remainder when is divided by How do we know what value to sub in ? If we are dividing by x - 2, we let x - 2 =0 and get x = 2. So we sub in 2 If we are dividing by x + 2, we let x + 2 =0 and get x = -2. So we sub in -2 If we are dividing by x - a, we let x - a =0 and get x = a. So we sub in a Question 1 Given that leaves a remainder of 6 when divided by and has a factor of , find the value of a and b. Next the question says that x + 2 is a factor Factor means that the remainder is zero Hence we can apply the remainder theorem By solving th

Question If the first 3 terms of the expansion of are , find the value of x and n Solution Recall the binomial expansion formula Next we need to simplify Going back to the question, We can then form 2 equations involving n and x Equation 1 Equation 2

Be careful of the negative signs when integrating sin and cos functions Question 1 a) Note the absence of the negative sign We divide by 2 as the differentiation of 2x is 2 b) Note the presence of the negative sign We divide by 2 as the differentiation of 2x is 2 Question 2 a) Note the presence of the negative sign b) Note the absence of the negative sign