Really Hard Math Question

Question

Given that f(x)=3x³-4x²+x+11, find g(x) if the roots of g(x) are:

a) three more than the roots of f(x)

b) a third the roots of f(x)

Show all works and explain work clearly.

Answer 1

f(x) = 3x^3 - 4x^2 + x + 11 is not Factorable by any convenient means, but if the curve has had all of its roots translated 3 to the right, the curve itself has been translated 3 to the right.

 

You do not need to factor the equation.  Replace the x in the equation with (x - 3), this translates the curve 3 to the right.

 

g(x) = 3(x - 3)^3 - 4(x - 3)^2 + (x - 3) + 11

g(x) = 3(x^3 -9x^2 + 27x -9) - 4(x^2 -6x +9) + x - 3 + 11

g(x) = 3x^3 - 27x^2 + 81x -27 - 4x^2 + 24x - 36 + x + 8

g(x) = 3x^3 - 31x^2 + 106x - 55

 

Again, if you wish to reduce all roots by two thirds, you replace the x in the equation with (3x)

 

g(x) = 3(3x)^3 - 4(3x)^2 + (3x) + 11

g(x) = 3(27x^3) - 4(9x^2) +3x + 11

g(x) = 81x^3 - 36x^2 + 3x + 11