Division of a Polynomial by a Polynomial

Steps for dividing a polynomial by a polynomial:

Step 1) Arrange the terms of the dividend and divisor in descending order of their degrees.
Step 2) Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.
Step 3) Multiply the divisor by the first term of the quotient and subtract the result from the dividend to obtain the remainder.
Step 4) Consider the remainder (if any) as dividend and repeat step (2) to obtain the second term of the quotient.
Step 5) Repeat the above process till we obtain a remainder which is either zero or a polynomial of degree less than that of the divisor.

Division of a Polynomial by a Polynomial Example1:

Divide \(x^3 + 1\) by (x + 1).
Solution:
Step 1) Set up in the form of long division in which the polynomials are arranged in descending order, leaving space for missing terms.
Step 2) Divide the first term of the dividend (\(x^3\)) by the first term of the divisor (x) and write the quotient above the line.
Step 3) Multiply the first term of the quotient (\(x^2\)) by each term of the divisor (x + 1) and write the product below the dividend.
Step 4) Subtract like terms and bring down one or more terms as needed.
Step 5) Now use the remainder \(-x^2\) as the new dividend and repeat Steps (2) to (4).
Step 6) Stop when the remainder becomes zero or when its degree becomes less than that of the divisor.

Therefore, \(x^3 + 1\) = (x + 1)(\({x^2}\) – x + 1).

Division of a Polynomial by a Polynomial Example2:

Divide \(2x^2 – 11x + 12\) by x – 4.
Solution:
Step 1) Set up in the form of long division in which the polynomials are arranged in descending order, leaving space for missing terms.
Step 2) Divide the first term of the dividend (\(2x^2\)) by the first term of the divisor (x) and write the quotient above the line.
Step 3) Multiply the first term of the quotient (2x) by each term of the divisor (x – 4) and write the product below the dividend.
Step 4) Subtract like terms and bring down one or more terms as needed.
Step 5) Now use the remainder -3x + 12 as the new dividend and repeat Steps (2) to (4).
Step 6) Stop when the remainder becomes zero or when its degree becomes less than that of the divisor.

Division Algorithm:

If a number is divided by another number then
Dividend = Divisor x Quotient + Remainder
Then, if 48 is divided by 5, then

Similarly, if a polynomial is divided by another polynomial, then
Dividend = Divisor x Quotient + Remainder
This is generally called the ‘division algorithm’.

Division of a Polynomial by a Polynomial Example3:

\(4z^4 + 7z^2 + 15\) = (\(4z^2 – 5\))(\(z^2 + 3\)) + 30.
The expression in the first parentheses of R.H.S. is the Divisor, expression in the second parentheses of R.H.S. is the Quotient and the remaining numeral is the Remainder.
Now we will check if division algorithm holds good or not.
Divisor x Quotient + Remainder = \(4z^2\)(\(z^2 + 3\)) -5(\(z^2 + 3\)) + 30
= \(4z^4 + 12z^2\) –\(5z^2 -15\) + 30
= \(4z^4 + 7z^2 + 15\) = Dividend.
So the division algorithm holds.

Division of a Polynomial by a Polynomial Example4:

Using division show that \(x^2 + 2 – 3x\) is a factor of \(2 – 3x^2 + x – x^3 + x^4\).
Solution:

As the remainder is 0 so \(x^2 + 2 – 3x\) is a factor of \(2 – 3x^2 + x – x^3 + x^4\). Therefore, \(2 – 3x^2 + x – x^3 + x^4\) = (\(x^2 + 2 – 3x\))(\(x^2 + 1 + 2x\)).

Division of a Polynomial by a Polynomial Example5:

Find whether \(4z^2 – 5\) is a factor of \(4z^4 + 7z^2 + 15\) or not.
Solution:

Remainder is 30. Since the remainder is not 0 so \(4z^2 – 5\) is not a factor of \(4z^4 + 7z^2 + 15\).

Division of a Polynomial by a Polynomial Example6:

Divide \(x^4 – 3x^3 – 13x^2 + 12x + 4\) by \(x^2 – x + 2\), and verify that Dividend = Divisor x Quotient + Remainder.
Solution:

The quotient = \(x^2\) – 2x – 17 and the remainder = –x + 38.
Verification:
Divisor x Quotient + Remainder = (\(x^2 – x + 2\))(\(x^2\) – 2x – 17) + (–x + 38)
= \(x^2\)(\(x^2\) – 2x – 17) – x(\(x^2\) – 2x – 17) + 2(\(x^2\) – 2x – 17) – x + 38
= \({x^4}\) – \({2x^3}\) – \({17x^2} – {x^3}\) + \({2x^2}\) + 17x + \({2x^2}\) – 4x – 34 – x + 38
= \({x^4}\) – \({2x^3}\) – \({x^3} – {17x^2}\)  + \({2x^2} + {2x^2}\) + 17x  – 4x – x – 34  + 38    { Collecting Like terms }
= \({x^4}\) – \({3x^3}\) – \({13x^2}\)  + 12x  + 4 = Dividend.
Hence verified.

Division of a Polynomial by a Polynomial Example7:

Divide \(x^3 – 8\) by (x – 2), x is not equal to 2.
Solution:

Therefore, Quotient = \(x^2 + 2x + 4\). Since the remainder is 0 so, (x – 2) is a factor of \(x^3 – 8\).

Division of a Polynomial by a Polynomial Example8:

What must be subtracted from \(x^4 + 6x^3 + 13x^2 + 13x + 8\) so that the resulting polynomial is exactly divisible by (\(x^2 + 3x + 2\)).
Solution:

Quotient = \(x^2 + 3x + 2\), Remainder = x + 4
So, if, we subtract the remainder x + 4 from \(x^4 + 6x^3 + 13x^2 + 13x + 8\), it will be exactly divisible by \(x^2 + 3x + 2\).