There we studied Definition, General form and Degree of Polynomial. Also we provided Links for study of Zeros, Simplified form, arranging and Adding of Polynomials.

That knowledge is a prerequisite here.

Here we deal with Multiplication and Dividing Polynomials.

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Solution to Example 1 of Dividing Polynomials : Let The two Polynomials be denoted by A and B. Then A = 5x^{2} + 3x - 7 and B = 2x^{2} - 4x + 9. AB = (5x^{2} + 3x - 7)(2x^{2} - 4x + 9) = 5x^{2}(2x^{2} - 4x + 9) + 3x(2x^{2} - 4x + 9) - 7(2x^{2} - 4x + 9) = 5x^{2}(2x^{2}) - 5x^{2}(4x) + 5x^{2}(9) + 3x(2x^{2}) + 3x(-4x) + 3x(9) -7(2x^{2}) -7(-4x) -7(9) = 10x^{4} - 20x^{3} + 45x^{2} + 6x^{3} - 12x^{2} + 27x - 14x^{2} + 28x - 63 = 10x^{4} + x^{3}(-20 + 6) + x^{2}(45 -12 -14) + x(27 + 28) - 63 = 10x^{4} - 14 x^{3} + 19x^{2} + 55x + 63 Ans.

The above method is horizontal method.

We can adopt column method here also.

In this method, we write multiplicand and the multiplier in descending powers of x, arrange one under another, and multiply the multiplicand by every term of the multiplier and add.

The degree of the product = 4 = the degree of the multiplicand (= 2) + the degree of the multiplier (= 2) We can observe that in multiplication of Polynomials, the degree of the product equals the sum of the degrees of themultiplicand and multiplier.

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step 1: Arrange the Polynomials in descending powers of the variable x. step 2: Divide the first term of the dividend by the first term of the divisor i.e. 10x^{2} ÷ 2x = 5x. Write the result as the term of the quotient. step 3: Multiply the term of the quotient i.e. 5x with each term of the divisor i.e. 2x - 3 giving 5x(2x - 3) = 10x^{2} - 15x and put the result under dividendfor subtracting. The subtraction gives 14x. step 4: Bring down one of the remaining terms of the dividend. This gives the remainder as 14x + 21. step 5: Now, use the remainder 14x + 21 as new dividend and repeat the steps 2 to 4. step 6: Stop when the remainder is zero or the degree of the remainder is less than the degree of the divisor. ∴ quotient = 5x + 7; remainder = 42. step 7: verification Check whether dividend = divisor x quotient + remainder is satisfied or not. Here divisor x quotient + remainder = (2x - 3)(5x + 7) + 42 = 10x^{2} + 14x - 15x - 21 + 42 = 10x^{2} - x +21 = dividend (verified.)

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