The division of polynomials p(x) and g(x) is expressed by the following “division algorithm” of algebra.
Let p(x) and g(x) be two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0. Then there exists unique polynomials q(x) and r (x)
such that
p(x) = g(x) q(x) + r (x) ... (1)
where r (x) = 0 or degree of r (x) < degree of g(x) .
The polynomial p(x) is the dividend, g(x) is the divisor, q(x) is the quotient and r (x) is the remainder.
(1) ==> Dividend = (Divisor x Quotient) + Remainder
Example 1 :
Divide the polynomial 2x^{3} - 6x^{2} + 5x + 4 by (x - 2)
Solution :
Let P(x) = 2x^{3} - 6x^{2} + 5x + 4 and g(x) = x - 2
To divide the given polynomial by x - 2, we have divide the first term of the polynomial P(x) by the first term of the polynomial g(x).
If we divide 2x^{3} by x, we get 2x^{2}. Now we have to multiply this 2x^{2} by x - 2. From this we get 2x^{3} - 4x^{2}.
Now we have to subtract 2x^{3} - 4x^{2} from the given polynomial. So we get -2x^{2} + 5x + 4.
Now we have to subtract 2x^{3} - 4x^{2} from the given polynomial. So we get -2x^{2} + 5x + 4.
repeat this process until we get the degree of p(x) ≥ degree of g(x).
So,
Quotient = 2x^{2} - 2x + 1
Remainder = 6
Example 2 :
Find the quotient and remainder when 4x^{3} - 5x^{2} + 6x - 2 by x - 1.
Solution :
So,
Quotient = 4x^{2} - x + 5
Remainder = 3
Example 3 :
Find the quotient and remainder when x^{3} - 7x^{2} - x + 6 by x + 2.
Solution :
So,
Quotient = x^{2} - 9x + 17
Remainder = -28
Example 4 :
Find the quotient and the remainder when 10- 4x + 3x^{2} is divided by x - 2.
Solution :
Let us first write the terms of each polynomial in descending order ( or ascending order).
Thus, the given problem becomes (10- 4x + 3x^{2}) ÷ (x - 2)
f(x) = 10- 4x + 3x^{2}
= 3x^{2} - 4x + 10
g(x) = x - 2
Step 1 :
In the first step, we are going to divide the first term of the dividend by the first first term of the divisor.
After changing the signs, +3x^{2} and -3x^{2} will get canceled. By simplifying, we get 2x + 10.
Step 2 :
In the second step again we are going to divide the first term that is 2x by the first term of divisor that is x.
So,
Quotient = 3x + 2
Remainder = 14
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