In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones.
If p(x) and g(x) are any two polynomials with g(x) is not equal to 0, then we can find polynomials q(x) and r(x) such that p(x)= g(x) × q(x)+r(x). Here, p(x) is dividend, g(x) is divisor, q(x) is quotient and r(x) is the remiander.
Steps to divide polynomials with help of example `p(x)= 4x+x^3+x^4-3x^2+5, g(x)= x^2+1-x`
1) Arrange terms of dividend and divisor in decreasing order of their degrees. Therefore, `p(x)= x^4+x^3-3x^2+4x+5, g(x)= x^2-x+1`
2) Then use the Euclid formula to divide. Here by solving this we get that dividend is `x^4+x^3-3x^2+4x+5,` divisor is `x^2-x+1`, quotient is `x^2+x-3` and remainder is 8
Shaalaa.com | Polynomials part 11 (Division Algorithm)
If the polynomial x4 – 6x3 + 16x2 – 25x + 10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a, find k and a.
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Find all the zeros of the polynomial 2x3 + x2 − 6x − 3, if two of its zeros are `-sqrt3` and `sqrt3`
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Verify division algorithm for the polynomial `f(x)= (8 + 20x + x^2 – 6x^3) by g(x) =( 2 + 5x –3x^2).`