Roots of polynomials

Question 1

If a cubic x^3 - 2x^2 + 3x - 4 = 0 has roots: a,b,c.

Find the equation of polynomials with roots: 2a, 2b, 2c.

Answer 1

Let w = 2x (as the new roots are double the original)

Hence x = w/2

Substitute in:

(w/2)^3 -2(w/2)^2 + 3(w/2) - 4 = 0

Multiple through by 8:

w^3 - 4w^2 + 12w - 32 = 0

Question 2

If a cubic x^3 - 2x^2 + 3x - 4 = 0 has roots a,b,c. Find the equation of the polynomial with roots a + 3, b + 3, c + 3:

Answer 2

Let w = x + 3 (as new roots are three bigger than original)

Hence x = w - 3

Substitute

(w-3)^3 - 2(w-3)^2 + 3(w-3) - 4 = 0

w^3 - 11w^2 + 42w - 58 = 0

Question 3

`What is the sum of the roots (letters)?

Answer 3

-b/a

Question 4

What is the sum of pairs of roots (letters)?

Answer 4

c/a

Question 5

What is the sum of triples of roots (letters)?

Answer 5

-d/a