## HOW TO FIND THE ROOTS OF A POLYNOMIAL OF DEGREE 4

Example 1 :

Solve the equation

x4+4x3+5x2+2x-2  =  0

which one root is -1 + i

Solution :

Let

f(x)  =  x4+4x3+5x2+2x-2

Since one of the root is complex number, the other root may be its conjugate.

So,  α = -1 + i  β = - 1 - i

By using these two roots we can find a quadratic equation which is the part of the original equation. First, let us find the quadratic equation.

General form of any quadratic equation:

x²-(α+β)x+αβ = 0

α  =  -1+i  β  =  -1-i

Sum of roots (α+β)  =  -1+i-1-i

= - 2

Product of roots (αβ)  =  (-1+i) (-1-i)

=  (-1)2 - i2

=  2

x2-(-2)x+2  =  0

x²+2x+2  =  0

This is the part of the equation of polynomial of degree 4.

x2+2x-1  =  0

we get other two roots.

a = 1, b = 2, c = -1

x  =  [- b ± √(b2-4 a c)]/2 a

x  =  [-2 ± √(22-4 (1)(-1))]/2 (1)

x  =  [- 2 ± √8]/2

x  =  [- 2 ± 2 √2]/2

x  =  -1±√2

Therefore the four roots are

-1 + i ,- 1 - i , - 1 + √2 , - 1 - √2.

Example 2 :

Solve the equation

6x4 − 35x3 + 62x2 − 35x + 6 = 0

Solution :

By suing synthetic division repeatedly, we may solve this problem.

2 and 3 are the solutions of the given polynomial. To know the other solutions, let us solve the quadratic equation.

6x2 - 5x + 1  =  0

6x2 - 2x - 3x + 1  =  0

2x (3x - 1) - 1(3x - 1)  =  0

(2x - 1) (3x - 1)  =  0

2x - 1  =  0 and 3x - 1  =  0

x  =  1/2  and x  =  1/3

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

Example 3 :

Solve

x4 + 3x3 - 3x - 1  =  0

Solution :

1 and -1 are the solutions of the given polynomial, to find other two solutions, let us solve the quadratic equation

x2 + 3x + 1  =  0

x = -b ± √(b2 - 4ac) / 2a

x = -3 ± √(9 - 4) / 2(1)

x = (-3 ± √5)/2

So, roots are -1, 1, (-3 ± √5)/2.

Example 4 :

Solve the equation 6x4 − 5x3 − 38x2 − 5x + 6 = 0 if it is known that 1/3 is a solution.

Solution :

1/3 and 3 are solutions of the given polynomial, to find other two solutions, let us solve the quadratic equation.

6x2 + 15x + 6  =  0

6x2 + 12x + 3x + 6  =  0

6x(x + 2) + 3(x + 2)  =  0

(6x + 3)(x + 2)  =  0

x  =  -1/2 and x = -2

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

Example 5 :

Solve the equation x4 −14x2 + 45 = 0

Solution :

Let x2  =  t

t2 − 14t + 45 = 0

t2 − 9t - 5t + 45 = 0

t(t - 9) - 5(t - 9)  =  0

(t - 5) (t - 9)  =  0

t  =  5 and t  =  9

x2  =  5, x2  =  9

x  =  ±√5, x = ±3

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

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

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6