ROOTS OF A CUBIC POLYNOMIAL WORKSHEET

Problem 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

Problem 2 :

Construct a cubic equations with roots 

i)  1, 2 and 3

ii)  1, 1 and -2

iii)  2, 1/2 and 1

Solution

Problem 3 :

If α, β and γ are the roots of the cubic equation

x³ + 2x² + 3x + 4 = 0

form a cubic equation whose roots are 

i)   2α, 2β and 2γ

ii)  1/α, 1/β and 1/γ

iii)  -α, -β and -γ

Solution

Problem 4 :

Solve the equation

3x³ - 16x² + 23x - 6 =0

if the product of two roots is 1.

Solution

Problem 5 :

Find the sum of the square of roots of the equation

2x⁴ - 8x³ + 6x² - 3 = 0

Solution

Problem 6 :

Solve the equation 

x³ - 9x² + 14x + 24 = 0

if it is given that two of its roots are in the ratio 3 : 2.

Solution

Problem 7 :

If  α, β and γ are the roots of the polynomial equation

ax³ + bx² + cx + d = 0

find the value of Σα/βγ in terms of the coefficients.

Solution

Problem 8 :

If  α, β, γ and δ are the roots of the polynomial equation

2x⁴ + 5x³ - 7x² + 8 = 0

find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and α β γ δ.

Solution

Problem 9 :

If p and q are the roots of equation lx² + nx + n = 0, show that √(p/q) + √(q/p) + √(n/l) = 0

Solution

Problem 10 :

If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must be equal to 

Solution

Problem 11 :

A 12 meter tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing.

Solution