 Hi and welcome to the session. Today we will discuss the following question. The question says give examples of polynomials p of x, g of x, q of x and r of x which satisfy the division algorithm and degree of q of x is equal to degree of r of x. Before proceeding for the solution let's recall division algorithm. It states that dividend is equal to divisor into quotient plus remainder. So if we have p of x as the dividend then p of x will be equal to divisor that is g of x into quotient that is q of x plus remainder that is r of x. So this is the key idea for this question. Now let's see the solution. Let us assume the polynomial p of x equal to x cube minus 3x square plus 5x plus 3 and the polynomial g of x equal to x square minus 2. So here as we can notice the degree of p of x is 3 degree of g of x is equal to 2. Now we need to find the values of q of x and r of x so that we will find by dividing the polynomial p of x by g of x. And then we will check that whether the four polynomials satisfy the division algorithm and the given condition that is degree of q of x is equal to degree of r of x or not. So let us divide x cube minus 3x square plus 5x plus 3 by x square minus 2. So first of all we will take the term x and multiplying the divisor that is x square minus 2 by x we will get x cube minus 2x. Now subtracting the terms we will get minus 3x square plus 7x plus 3. Now let's take minus 3 so multiplying the divisor by minus 3 we will get minus 3x square plus 6. Now subtracting the terms we will get 7x minus 3. So here quotient that is q of x is equal to x minus 3 and remainder that is r of x is equal to 7x minus 3. So this implies degree of q of x is equal to 1. Degree of r of x is also equal to 1. This implies the degree of q of x is equal to degree of r of x that means the four polynomials satisfy the given condition that is degree of q of x is equal to degree of r of x. Now let's see whether they satisfy the division algorithm or not. Now we know the division algorithm. So let's start with the r h s that is g of x into q of x plus r of x will be equal to here g of x is x square minus 2 into q of x that is x minus 3 plus r of x that is 7x minus 3. So let's multiply the first two polynomials and this will give us x cubed minus 3x square minus 2x plus 6. Now this we will write as it is plus 7x minus 3. So on solving this we will get x cubed minus 3x square now minus 2x plus 7x is plus 5x and plus 6 minus 3 is plus 3. Now x cubed minus 3x square plus 5x plus 3 is same as the px. So this is equal to px that means the four polynomials p of x g of x q of x r of x satisfy the division algorithm. So these four polynomials satisfying both the conditions that is division algorithm and second degree of q of x is equal to degree of r of x that means p of x equal to x cubed minus 3x square plus 5x plus 3 g of x equal to x square minus 2 q of x equal to x minus 3 and r of x equal to 7x minus 3 is the required answer to this question. Please note that there can be several examples for this question which will satisfy both of these conditions. So with this we finish this session hope you must have understood the question. Goodbye and have a nice day.