We will learn Division Algorithm for Polynomials. Now, let us understand the text book definition first. It is given in the book, the statement is, if p of x and g of x are any two polynomials with g of x not equal to zero, then we can find polynomials q of x and r of x such that, p of x is equal to g of x into q of x plus r of x, where r of x is equal to zero or degree of r of x is less than degree of g of x. This is a book definition. Let us understand it properly.
The Division Algorithm for Polynomial may be written as Dividend is equal to Divisor into Quotient plus Remainder. For example, let us do a division where the dividend is 10, divisor is 5. When you divide 10 by 5, the quotient is 2 because 5 into 2 is 10. And 5 into 2, 10, remainder will be, how much. You subtract 10 and 10, you will get zero. And whenever remainder is zero, what we say, 5 is a factor of 10. So, what we understand if the remainder r of x is equal to zero then the polynomial g of x is the factor of the polynomial p of x, okay. Where the g of x is a divisor, p of x is a dividend, q of x is quotient and r of x is the remainder. So, in simple words it is nothing but dividend is equal to divisor into quotient plus remainder.
Exercise: Divide the polynomial p of x by the polynomial g of x and find the quotient and the remainder in each of the following. So, they have given the two polynomials and we have to divide that.
Solution: First I will copy the two polynomials, first polynomial is x cube minus 3x square plus 5x minus 3, and then I am copying the second polynomial which is x square minus 2. The dividend should be in the index form. Okay, the first polynomial is the dividend and it should be in the index form. What do you mean by index form? Index form means all the powers of the variable are present in the descending order. Descending order means highest power, if you see in the first one it is 3, it should be in descending power that means all the powers should be there. That means 3, 2, 1 and even zero should be there that is called descending powers, okay. Is this dividend in the descending order of index, yes, it is in the descending order of index because all the powers are there. All the powers of the variables are present in the descending order. So, what we will do, we will perform the division, first inside we will write the dividend which is the first polynomial as it is. X cube minus 3x square plus 5x minus 3 and outside we will write the divisor. Divisor is x square minus 2. Now let us perform this division. So, let us go to our working column and perform the division. How to perform the division? The first term of dividend is divided by the first term of a divisor. Which is the first term of dividend? First term of the dividend is x cube, okay. So, x cube is divided by first term of the divisor. Which is the first term of the divisor? X square. So first term of dividend is divided by first term of divisor. And if you do this division, let us divide. If you do this division, what will we get? We will get x, now this x becomes the quotient, okay. This x is now multiplied with the entire divisor, okay. X is multiplied with the divisor. If you multiple x with the divisor. What is the divisor? The divisor is x square minus 2. If you multiple this x with the divisor x square minus 2, what you will get? Open the brackets, you will get x into x square will be x cube and x into 2 will be minus 2x, okay. Now, this becomes the result. This result we will write it below the dividend. So, below the x cube we will write x cube and below the x term we will write minus 2x. So, x cube and below the x term we are writing minus 2x. Let us now do the subtraction and whenever we are doing the subtractions the signs will change plus becomes minus and the minus becomes plus, okay. So, x cube and minus x cube will get cancelled, okay. This minus 3x square comes down as it is, carried down as it is. Plus 5x plus 2x becomes plus7x and minus 3 is carried down, okay. Then how to divide further? Again the same procedure, first term of the dividend is divided by the first term of the divisor. What is the first term of the dividend? Minus 3x square is divided by the first term of the divisor, what is the first term of the divisor, x square. So, let us do the division. X square x square gets cancelled, when x square x square gets cancelled what is the result we get. The result we get is minus 3. This minus 3, you write it in the quotient. Let us write minus 3 in the quotient. Now this minus 3 is multiplied with the entire divisor, what is the divisor? X square minus 2, so minus 3 is multiplied with the divisor x square minus 2. If you open the bracket, what you will get minus 3 into x square into minus 3x square, minus 3 into minus 2 becomes plus 6. This we will write in the dividend. Below the x square we will write the x square term and below the constant we will write the constant term. So, minus 3x square written below the minus 3x square and the plus 6 we will write it below the constant term which is minus 3, okay. Now, while subtraction, the signs change of the second polynomial what we wrote, minus becomes plus and plus becomes minus. So, minus 3x square plus 3x square gets cancelled, okay. The 7x gets carried down as it is. And minus 3 minus 6 becomes minus 9, okay. Now what do you observe here. The degree of the remainder that we got is less than the degree of the divisor. Because 7x minus 9 the degree is 1 and the degree of the divisor, it is x square minus 2, it is 2. The degree of the remainder is less than the degree of divisor, so what we do? We will stop the division. And since we are stopping the division what happens, this x minus 3 what we got that becomes the quotient. This 7x minus 9, what we got, becomes the remainder. So, what is the quotient x minus 3, what is the remainder 7x minus 9. That is the result.