CONSTRUCTING A CUBIC EQUATION WITH GIVEN ROOTS

Constructing a Cubic Equation with Given Roots :

In this section, you will learn how to construct a cubic equation with the roots given.

Constructing a Cubic Equation with Given Roots - Practice Questions

Question 1 :

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 be the side length of the cube 

Volume of cube is

=  x³

Length of cuboid  =  (x + 1)

Breadth of cuboid  =  (x + 2)

Height of cuboid  =  (x + 3)

Volume of cuboid  =  (x + 1) (x + 2) (x + 3)

(x + 1) (x + 2) (x + 3)  =  52

(x² + 3x + 2) (x + 3)  =  52

x³ + 3x² + 3x² + 9x + 2x + 6  =  x³ + 52

x³ - x³ + 6x² + 11x + 6 - 52  =  0

6x² + 11x - 46  =  0

6x² - 12x + 23x  - 46  =  0

6x (x - 2) + 23(x - 2)  =  0

6x + 23  =  0, x - 2  =  0

x = 2

Volume of cube is

=  x³

=  2³

=  8 cubic units.

Question 2 :

Construct a cubic equation with roots.

(i) 1, 2 and 3 (ii) 1,1, and −2 (iii) 2, 1/2 and 1.

Solution :

(i) 1, 2 and 3 

x = 1, x = 2 and x = 3

(x - 1) (x - 2) (x - 3)  =  0

(x² - 2x - x + 2)(x - 3)  =  0

(x² - 3x + 2)(x - 3)  =  0

x³ - 3x² - 3x² + 9x + 2x - 6  =  0

x³ - 6x² + 11x - 6  =  0

(ii) 1, 1 and −2

x = 1, x = 1 and x = -2

(x - 1) (x - 1) (x + 2)  =  0

(x² - x - x + 1)(x + 2)  =  0

(x² - 2x + 1)(x + 2)  =  0

x³ + 2x² - 2x² - 4x + 1x + 2  =  0

x³  - 3x + 2  =  0

(iii) 1, 1/2 and 1

x = 1, x = 1/2 and x = 1

(x - 1) (x - 1/2) (x - 1)  =  0

(x - 1)²(x - 1/2)  =  0

(x² - 2x + 1) (2x - 1)/2)  =  0

2x³ - x² - 4x² + 2x + 2x - 1  =  0

2x³ - 5x² + 4x - 1  =  0

After having gone through the stuff given above, we hope that the students would have understood how to construct a cubic equation with the roots given. 

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

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.