**Constructing a Cubic Equation with Given Roots :**

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

**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

= x^{3}

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

(x^{2} + 3x + 2) (x + 3) = 52

x^{3} + 3x^{2} + 3x^{2} + 9x + 2x + 6 = x^{3} + 52

x^{3 }- x^{3} + 6x^{2} + 11x + 6 - 52 = 0

6x^{2} + 11x - 46 = 0

6x^{2} - 12x + 23x - 46 = 0

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

6x + 23 = 0, x - 2 = 0

x = 2

Volume of cube is

= x^{3}

= 2^{3}

= 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

(x^{2} - 2x - x + 2)(x - 3) = 0

(x^{2} - 3x + 2)(x - 3) = 0

x^{3} - 3x^{2} - 3x^{2} + 9x + 2x - 6 = 0

x^{3} - 6x^{2} + 11x - 6 = 0

(ii) 1, 1 and −2

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

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

(x^{2} - x - x + 1)(x + 2) = 0

(x^{2} - 2x + 1)(x + 2) = 0

x^{3} + 2x^{2} - 2x^{2} - 4x + 1x + 2 = 0

x^{3} - 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

(x^{2} - 2x + 1) (2x - 1)/2) = 0

2x^{3} - x^{2} - 4x^{2} + 2x + 2x - 1 = 0

2x^{3} - 5x^{2} + 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.

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