The question is as follows:
Suppose $g(x)$ is a polynomial of degree $5$. $g(0)=2$ and $g(1)=−623/15$ . Moreover, suppose $g(x)$ has local extreme values at $2√2$,$−2√2$,$√6$ and $√-6$ . Complete this sentence: The formula for this polynomial is $g(x)=$
I can do this quite easily for a quadratic, which is all that we have done so far, however I am not sure how to do this for a fifth degree polynomial. I tried doing it the same way and it became incredibly disorganized, so I'm not sure if that's how you are supposed to do it, or if I am missing something.
Hint: let $g(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$. Then $$g'(x)=5a_5x^4+4a_4x^3+3a_3x^2+2a_2x+a_1= \\5a_5(x-2\sqrt2)(x+2\sqrt2)(x-\sqrt 6)(x+\sqrt 6) = \\5a_5(x^2-8)(x^2-6)$$ From the equality of polynomials you should be able to create 4 equations for $a_1, a_2, a_3, a_4, a_5$. In fact, you can see that $a_4=a_2=0$ right away. The remaining two conditions are to create two more equations.