Solved Problem on Division Algorithm for Polynomials

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. 

Question 1: Find all the zeros of the polynomial f(x) = 2x³ -5x² -4x + 3 if one of roots is . 

Solution: 

x =  is a root of the polynomial (Given) 

Now we know from the fact stated above, (x – ) is a factor of the given polynomial. So, for finding out the other zeros, we need to divide the polynomial with this factor. 

So we get 2x² -4x – 6 as quotient. 

The remaining two roots are roots of this polynomial. 

2x² – 4x – 6 = 0

⇒ 2x² -6x + 2x – 6 = 0 

⇒ 2x(x – 3) + 2 (x – 3) = 0 

⇒ (2x + 2) (x – 3) = 0 

x = -1 and x = 3

Thus, the remaining two roots are x = -1 and x = 3. 

Question 2: Divide the polynomial 5x⁴ -3x³ + 2x² – 1 with x³ – 1. 

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. 

Solution: 

We are given two zeros of the polynomial. We know that, x – √2 and x + √2 are the factors of the polynomial. 

Two find the other roots let’s divide the polynomial with both of these. 

(x – √2)(x + √2) 

 = x² – 2

Dividing the polynomial with x² – 2. 

The quotient polynomial is given by 2x² – 3x + 1

The remaining two roots are also the roots of this polynomial. 

2x² – 3x + 1

⇒ 2x² – 2x -x + 1

⇒ 2x(x -1) -1(x – 1) 

⇒ (2x – 1) (x – 1) = 0

So, the roots come out to be x = and x= 1. 

Thus, all the roots are x = 1, √2, -√2 and 

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).