# POLYNOMIAL EQUATIONS

### Features of Roots of Polynomial Equations Software:

The Roots of Polynomial Equations is a mathematics educational software, which facilitates to solve (find all the roots of) a given polynomial equation. It can find roots of polynomial equations upto fourth order. (ie., Quartic Equations). This software can find both real and complex number roots.

• Open Genius Maker software and click "Roots of Polynomial" button. It opens the Roots of Polynomial window.
• You may see 2 menus, the first one for selecting the order of polynomial and the second one display of input and output fields.

#### Example - 1

• Find the roots of the quadratic equation x² + 3 x - 10 = 0
• Click "Clear Data" button.

• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x² + B x + C = 0

• In the current problem, A = 1, B = 3 and C = -10.

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

Two Real roots
x = 2
x = -5

• Hence the roots of the given quadratic equation are 2 and -5.

#### Example - 2

• Find the roots of the quadratic equation x² - 6 x + 13 = 0
• Click "Clear Data" button.

• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x² + B x + C = 0

• In the current problem, A = 1, B = -6 and C = 13.

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

Two Imaginary roots
x = ( 3 + 2 i )
x = ( 3 - 2 i )

• Hence the roots of the given quadratic equation are (3 + 2i) and  (3 - 2i) .

#### Example - 3

• Find the roots of the cubic equation x² - 10 x + 25 = 0
• Click "Clear Data" button.

• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x² + B x + C = 0

• In the current problem, A = 1, B = -10 and C = 25.

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

Two Real roots, which are equal
x = 5
x = 5

• Hence the 2 roots of the given quadratic equation are same and are equal to 5 .

#### Example - 4

• Find the roots of the cubic equation x³ - 3 x² - 4 x + 12 = 0
• Click "Clear Data" button.

• Select the third option, "Cubic equation" from the top menu.

• The title of the second menu should read as "Cubic equation"
• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x³ + B x² + C x + D = 0

• In the current problem, A = 1, B = -3, C = -4 and D = 12

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

Three real roots
x = 3
x = -2
x = 2

• Hence the roots of the given cubic equation are 3, -2 and 2.

#### Example - 5

• Find the roots of the cubic equation x³ - 14 x² + 69 x - 116 = 0
• Click "Clear Data" button.

• Select the third option, "Cubic equation" from the top menu.

• The title of the second menu should read as "Cubic equation"
• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x³ + B x² + C x + D = 0

• In the current problem, A = 1, B = -14, C = 69 and D = -116

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

One Real root and Two Imaginary roots
x = 4
x = ( 5 + 2 i )
x = ( 5 - 2 i )

• Hence the roots of the given cubic equation are 4, (5 + 2i) and (5 - 2i).

#### Example - 6

• Find the roots of the polynomial quartic equation x4 - 3 x³ + 40 x² - 26 x - 600 = 0
• Click "Clear Data" button.

• Select the fourth option, "Quartic equation" from the top menu.

• The title of the second menu should read as "Quartic equation"
• Now you can enter the values of co-efficient in the equation.
• You may note the format displayed for input. In this case, it reads as

A x4 + B x³ + C x² + D x + E = 0

• In the current problem, A = 1, B = -3, C = 40, D = -26 and E = -600

• Enter the values in the input boxes accordingly.
• Click "Find All Roots" button.
• You can see the result appearing as shown below.

Two Real roots and Two Imaginary roots
x = 4
x = -3
x = ( 1 + 7 i )
x = ( 1 - 7 i )

• Hence the roots of the given Quartic equation are 4, 3, (1 + 7i) and (1 - 7i).

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