**Find the Other Roots of the Polynomial Equation of Degree 6 :**

Here we are going to see some example problems of solving polynomial of degree 6.

**Question 1 :**

Find all zeros of the polynomial x^{6} − 3x^{5} − 5x^{4} + 22x^{3} − 39x^{2} − 39x + 135, if it is known that 1 + 2i and √3 are two of its zeros.

**Solution :**

The four roots are 1 + 2i, 1 - 2i, √3 and -√3.

Quadratic equation whose roots are 1 + 2i and 1 - 2i.

Sum of roots = 2

Product of roots = 1 + 4 = 5

x^{2} - 2x + 5

Sum of roots = 0

Product of roots = -3

(x^{2} - 3) is a factor.

(x^{2} - 2x + 5) (x^{2} - 3) = x^{4} - 3x^{2} - 2x^{3} + 6x + 5x^{2} - 15

= x^{4} - 2x^{3} + 2x^{2} + 6x - 15

x^{2} - x - 9 = 0

x = (-b ± √b^{2} - 4ac)/2a

x = (1 ± √1 + 36)/2

x = (1 ± √37)/2

Hence the zeroes are 1 + 2i, 1 - 2i, √3, -√3, (1 ± √37)/2.

**Question 2 :**

Solve the cubic equation

(i) 2x^{3} − 9x^{2} +10x = 3

**Solution :**

2x^{3} − 9x^{2} +10x - 3 = 0

(x -1) is a factor

= 2x^{2} - 7x + 3

= 2x^{2} - 1x - 6x + 3

= x(2x - 1) - 3(2x - 1)

= (x - 3) (2x - 1)

(x - 1) (x - 3) (2x - 1) = 0

x - 1 = 0, x - 3 = 0, 2x - 1 = 0

x = 1, x = 3, x = 1/2

Hence the solutions are 1, 3 and 1/2.

(ii) 8x^{3} − 2x^{2} − 7x + 3 = 0.

**Solution :**

(x + 1) is a factor.

The other factors are 8x^{2} - 10x + 3

= 8x^{2} - 12x + 2x + 3

= 4x(2x - 3) -1(2x - 3)

= (4x - 1) (2x - 3)

By equating them to zero, we get

x + 1 = 0, 4x - 1 = 0, 2x - 3 = 0

x = -1, x = 1/4, x = 3/2

Hence the solutions are -1, 1/4 and 3/2.

**Question 3 :**

Solve the equation x^{4} −14x^{2} + 45 = 0

**Solution :**

Let x^{2} = t

t^{2} − 14t + 45 = 0

t^{2} − 9t - 5t + 45 = 0

t(t - 9) - 5(t - 9) = 0

(t - 5) (t - 9) = 0

t = 5 and t = 9

x^{2 }= 5, x^{2 }= 9

x = ±√5, x = ±3

Hence the solutions are √5, -√5, 3 and -3.

After having gone through the stuff given above, we hope that the students would have understood, "Find the Other Roots of the Polynomial Equation of Degree 6".

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