5th degree polynomial question5

In this page 5th degree polynomial question5 we are going to see how to solve the polynomial which is having degree 5.

Question 5 :

Solve  6 x⁵ + 11 x⁴ - 33 x³ - 33 x² + 11 x + 6

Solution :

The degree of this equation is 5. Therefore we can say there will be 5 roots for this equation.

This is a reciprocal equation of odd degree with like terms. So -1 is one of the root of this equation. The other roots are given by

6x⁴ + 5 x³ - 38 x² + 5 x + 6 = 0

Dividing the entire equation by x²

6x⁴/x² + 5 x³/x² - 38 x²/x² + 5 x/x² + 6/x² = 0

6 x² + 5 x - 38 +  (5/x) + (6/x²) = 0

6 x² + (6/x²) + 5 x +  (5/x) - 38 = 0

6 (x² + 1/x²) + 5 (x + 1/x) - 38 = 0 ------ (1)

Let x + 1/x = y

To find the value of x² + 1/x² from this we have to take squares on both sides

(x + 1/x)² = y²

x² + 1/x² + 2 x (1/x) = y²

x² + 1/x² + 2 = y²

x² + 1/x² = y² - 2

So we have to plug y² - 2 instead of x² + 1/x²

Let us plug this value in the first equation

6 (y² - 2) + 5 y - 38 = 0

6 y² - 12 + 5 y - 38 = 0

6 y² + 5 y - 12 - 38 = 0

6 y² + 5 y - 50 = 0

6 y² + 20 y - 15 y - 50 = 0

2 y (3y + 10) - 5 (3y + 10) = 0

(2y - 5) (3y + 10) = 0

2y - 5 = 0

2y = 5

y = 5/2

y = 5/2

3y + 10 = 0

3y = -10

y = -10/3

y = -10/3

x + 1/x = y

(x² + 1)/x = 5/2

2(x² + 1) = 5 x

2x² + 2 - 5x = 0

2x² - 5x + 2  = 0

2x² - 4 x - 1 x + 2  = 0

2x(x - 2) - 1(x - 2)  = 0

(2x - 1) (x - 2)  = 0

2 x - 1 = 0             x - 2 = 0

2x = 1                    x = 2

x = 1/2

x + 1/x = y

(x² + 1)/x = -10/3

3(x² + 1) = -10 x

3x² + 3 + 10x = 0

3x² + 10x + 3  = 0

3x² + 1 x + 9 x + 3  = 0

x(3x + 1) + 3(3x + 1)  = 0

(3x + 1) (x + 3)  = 0

3 x + 1 = 0             x + 3 = 0

3x = -1                    x = -3

x = -1/3

x = -1/3
5th degree polynomial question5

Therefore the 5 roots are x = -3,-1/3,2,1/2,-1

This is the example problem in the topic solving polynomial of degree5. You can try the following sample test to understand this topic much better.

 Questions Solution Question 1 :Solve 6 x⁵ - x⁴ - 43 x³ + 43 x² + x - 6 Solution Question 2 :Solve 8 x⁵ - 22 x⁴ - 55 x³ + 55 x² + 22 x - 8 Solution Question 3 :Solve  x⁵ - 5 x⁴ + 9 x³ - 9 x² + 5 x - 1 Solution Question 4 :Solve  x⁵ - 5 x³ + 5 x² - 1 Solution

5th Degree Polynomial Question5 to Algebra 