Division Algorithm for Polynomials
Polynomials are those algebraic expressions that contain variables, coefficients, and constants. For Instance, in the polynomial 8x2 + 3z – 7, in this polynomial, 8,3 are the coefficients, x and z are the variables, and 7 is the constant. Just as simple Mathematical operations are applied on numbers, these operations can also be applied on different polynomials, applying different operations on polynomials gives a new polynomial, say p(x) is a polynomial multiplied with q(x), then, the new polynomial g(x) = p(x) × q(x).
Division Algorithm for Polynomials
Division algorithm states that,Â
If p(x) and g(x) are two polynomials with g(x) ≠0, then we can find polynomials q(x) and r(x) such that,Â
p(x) = g(x) x g(x) + r(x)Â
Where r(x) = 0 or degree of r(x) < degree of g(x)Â
Dividend = Quotient × Divisor + RemainderÂ
Let’s look at some steps for doing this kind of division and then solve some examples related to it.Â
- In this step, arrange the divisor and dividend in an order which is decreasing according to their degrees.
- First-term of the Quotient is given by dividing the highest degree term of the dividend by the highest degree term of the divisor.Â
- The second term of the quotient is obtained by dividing the highest degree term of the new dividend obtained as the remainder by the highest degree term of the divisor.
- Continue this process till the degree of remainder is less than the degree of divisor.
Let’s look at some sample problems related to this algorithm.
Question 1: Divide the polynomial x3 + x2 – 1 with x – 1.Â
Solution:Â
We just need to follow the same steps as mentioned above.Â
So, the quotient here is x2 + 2x + 2 and remainder is 1.Â
Question 2: Divide the polynomial x4 + x3 + x2 – 1 with x3 – 1.Â
Solution:
So, the quotient comes out to be x + 1 and the remainder x2 + x.Â
Use of Division Algorithm in finding the zeros of a polynomial
Suppose we have a polynomial P(x) = 0 of degree 3. If we are given a root x = r of that polynomial. We can find the other two roots by dividing the polynomial with (x -r). Let’s see it with an example.Â
Question 1: Find all the zeros of the polynomial f(x) = 2x3 -5x2 -4x + 3 if one of roots is .Â
Solution:Â
x =Â Â is a root of the polynomial (Given)Â
Now we know from the fact stated above, (x –Â ) is a factor of the given polynomial. So, for finding out the other zeros, we need to divide the polynomial with this factor.Â
So we get 2x2 -4x – 6 as quotient.Â
The remaining two roots are roots of this polynomial.Â
2x2 – 4x – 6 = 0
⇒ 2x2 -6x + 2x – 6 = 0Â
⇒ 2x(x – 3) + 2 (x – 3) = 0Â
⇒ (2x + 2) (x – 3) = 0Â
x = -1 and x = 3
Thus, the remaining two roots are x = -1 and x = 3.Â
Question 2: Divide the polynomial 5x4 -3x3 + 2x2 – 1 with x3 – 1.Â
Solution:
The remainder is 3 and quotient is 5x3 + 2x2 + 4x + 4
Question 3: Find all the zeros of 2x4 – 3x3 -3x2 + 6x – 2. We know that two zeros are √2 and -√2.Â
Solution:Â
We are given two zeros of the polynomial. We know that, x – √2 and x + √2 are the factors of the polynomial.Â
Two find the other roots let’s divide the polynomial with both of these.Â
(x – √2)(x + √2)Â
 = x2 – 2
Dividing the polynomial with x2 – 2.Â
The quotient polynomial is given by 2x2 – 3x + 1
The remaining two roots are also the roots of this polynomial.Â
2x2 – 3x + 1
⇒ 2x2 – 2x -x + 1
⇒ 2x(x -1) -1(x – 1)Â
⇒ (2x – 1) (x – 1) = 0
So, the roots come out to be x =Â and x= 1.Â
Thus, all the roots are x = 1, √2, -√2 andÂ
Last Updated :
19 Mar, 2024
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