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Please study

Polynomial before Dividing Polynomials

if you have not already done so.

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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Dividing Polynomials

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Multiplication :

While multiplying two Polynomials, we use distributive laws.

Example 1 of Dividing Polynomials :

Multiply the Polynomials in Example 1 of Adding Polynomials.

Solution to Example 1 of Dividing Polynomials :
Let The two Polynomials be denoted by A and B.
Then A = 5x2 + 3x - 7 and B = 2x2 - 4x + 9.
AB = (5x2 + 3x - 7)(2x2 - 4x + 9)
= 5x2(2x2 - 4x + 9) + 3x(2x2 - 4x + 9) - 7(2x2 - 4x + 9)
= 5x2(2x2) - 5x2(4x) + 5x2(9) + 3x(2x2) + 3x(-4x)
+ 3x(9) -7(2x2) -7(-4x) -7(9)
= 10x4 - 20x3 + 45x2 + 6x3 - 12x2 + 27x - 14x2 + 28x - 63
= 10x4 + x3(-20 + 6) + x2(45 -12 -14) + x(27 + 28) - 63
= 10x4 - 14 x3 + 19x2 + 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.

5x2 + 3x - 7
2x2 - 4x + 9
10x4 + 6x3 - 14x2
                 - 20x3 - 12x2 + 28x
                               + 45x2 + 27x - 63
                      10x4 - 14x3 + 19x2 + 55x - 63 Ans.

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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Dividing Polynomials :

Let us see the steps in the procedure of
Dividing Polynomials by means of an example.

Example 2 of Dividing Polynomials :

Divide 10x2 - x + 21 by 2x - 3

Solution to Example 2 of Dividing Polynomials :

          divisor 2x - 3)10x2 -     x + 21 ( 5x + 7 quotient
10x2 - 15x
             +14x + 21
             +14x - 21
                                + 42 remainder

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. 10x2 ÷ 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) = 10x2 - 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
= 10x2 + 14x - 15x - 21 + 42
= 10x2 - x +21 = dividend (verified.)

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