I need answers to my homework plz!

Question

1.) Timmy solved the polynomial equation x^5-x^4-2x^2-24x+24=0 and his solutions were x=-1,(square root) 6, 2i. Explain his error(s) and list the correct solutions below.

2.) Apply the function transformation of g(x)=f(x-2)+4 to f(x)=x^3-8. Describe the transformation and find the final form of the function after the transformation.

Answer 1

For 1), if we know some theory about polynomials, we can immediately see that the sqrt(6) cannot be a root. This is because (due to Galois), the roots of a fifth or higher degree polynomial cannot be expressed in terms of roots. We also know immediately that 2i cannot be a root. Complex roots of a polynomials with real coefficients always come in pairs. So, if 2i were a root, we would also have -2i being a root. But, because the polynomial is not an odd function, 2i and -2i cannot both be roots.

I used wolfram alpha to list out the roots, and they all appear to be irrational. So I don't know how you would be expected to calculate the roots yourself. Perhaps you made a typo in writing the polynomial.

For 2), we have g(x) = (x-2)^3-8+4 = (x-2)^3-4. Here, we have transformed the graph of f by shifting right by 2 units, and down by 4 units.

Hope this helps.