HOW TO MAKE A CUBIC POLYNOMIAL WHEN ROOTS ARE GIVEN

Construct a cubic equations with roots 

Problem 1 :

1, 2 and 3

Solution :

α = 1, β = 2 and γ = 3

Coefficient of x² :

-(α + β + γ) =  -(1 + 2 + 3) ==> -6

Coefficient of x :

α β + βγ + γα = 1(2) + 2(3) + 3(1)

= 2 + 6 + 3

= 11

Constant term :

-αβγ = -1(2)(3)

= -6

ax³ + bx² + cx + d = 0

x³ - 6x² + 11x - 6 = 0

Problem 2 :

1, 1 and -2

Solution :

α = 1, β = 1 and γ = -2

Coefficient of x² :

-(α + β + γ) =  -(1 + 1 - 2) ==> 0

Coefficient of x :

α β + βγ + γα = 1(1) + 1(-2) + (-2)(1)

= 1 - 2 - 2

= -3

Constant term :

-αβγ = -[1(1)(-2)]

= 2

ax³ + bx² + cx + d = 0

x³ + 0x² - 3x + 2 = 0

x³ - 3x + 2 = 0

Problem 3 :

2, 1/2 and 1

Solution :

α = 2, β = 1/2 and γ = 1

Coefficient of x² :

-(α + β + γ) =  -(2 + 1/2 + 1) ==> -7/2

Coefficient of x :

α β + βγ + γα = 2(1/2) + (1/2)(1) + (1)(2)

= 1 + (1/2) + 2

= 7/2

Constant term :

-αβγ = -[2(1/2)(1)]

= -1

ax³ + bx² + cx + d = 0

x³ + (-7/2)x² + (7/2)x - 1 = 0

2x³ - 7x² + 7x - 2 = 0

Problem 4 :

If the sides of a cubic box are  increased by 1, 2, 3 units respectively to form a cuboid then the volume is increased by 52 cubic units. Find the volume of the cuboid.

Solution :

Let x side of cubic box before increasing.

After increasing each side, the new lengths will be x + 1, x + 2 and x + 3.

Volume of cubic box = x³ + 52

length x width x height = x³ + 52

(x + 1)(x + 2)(x + 3) = x³ + 52

(x + 1)(x² + 5x + 6) = x³ + 52

5x² + 6x + x² + 5x + 6 - 52 = 0

 6x² + 11x - 46 = 0

 6x² - 12x + 23x - 46 = 0

6x(x - 2) + 23(x - 2) = 0

(6x + 23) (x - 2) = 0

x = 2 and x = -23/6

Volume of cubic box = x³ + 52

= 2³ + 52

= 8 + 52

= 60 cubic units.