Solving polynomial of degree5 :

Here we are going to see how to solve the polynomial which is having degree 5.

**Example 1 :**

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

6 x⁴ + 5 x³ - 38 x² + 5 x + 6 = 0

Dividing the entire equation by x²

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

6 x² + 5 x - 38 + 5 (1/x) + 6(1/x²) = 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 - 15 y + 20 y - 50 = 0

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

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

3y + 10 = 0

3 y = -10

**y = -10/3**

2 y - 5 = 0

2 y = 5

**y = 5/2**

x + 1/x = y

(x² + 1)/x = 5/2

2(x² + 1) = 5 x

2x² + 2 - 5x = 0

2x² - 5x + 2 = 0

2x² - 4x - 1x + 2 = 0

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

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

2x - 1 = 0 x - 2 = 0

2 x = 1 **x = 2**

** x = 1/2 **

x + 1/x = y

(x² + 1)/x = -10/3

3(x² + 1) = -10x

3x² + 3 = -10 x

3x² + 10 x + 3 = 0

3x² + 9 x + 1 x + 3 = 0

3x (x + 3) + 1(x + 3) = 0

(3x + 1) = 0 (x + 3) = 0

3x = -1 **x = -3**

** x = -1/3**

Hence the 5 roots are **x = -1/3,-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.

(1) Solve 6 x⁵ - x⁴ - 43 x³ + 43 x² + x - 6 Solution

(2) Solve 8 x⁵ - 22 x⁴ - 55 x³ + 55 x² + 22 x - 8

(3) Solve x⁵ - 5 x⁴ + 9 x³ - 9 x² + 5 x - 1

(4) Solve x⁵ - 5 x³ + 5 x² - 1

(5) Solve 6 x⁵ + 11 x⁴ - 33 x³ - 33 x² + 11 x + 6

After having gone through the stuff given above, we hope that the students would have understood "Solving polynomial of degree5"

Apart from the stuff given above,
if you want to know more about "Solving polynomial of degree5", __please click here.__

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**