1. ## Polynomial Shift

Suppose P(x) = anxn + an-1xn-1+an-2xn-2+…+a1x1+a0 is any polynomial of degree 1 or higher with real coefficients. Show that there must exist a real number k so that if we create the new polynomial Q(x) = P(x-k), then Q(x) = bnxn + bn-2xn-2 + … + b1x1 + b0 meaning that the shift has caused the new polynomial to lose its x^(n-1) term.

I know it has something to do with the significance of the 2nd coefficient polynomial, but I get lost after that. any help appreciated.

2. Originally Posted by amma0913

Suppose $\displaystyle P(x) = a_nx^n + a_{n-1}x^{n-1}+a_{n-2}x^{n-2} + \cdots +a_1x + a_0$ is any polynomial of degree 1 or higher with real coefficients. Show that there must exist a real number $\displaystyle k$ so that if we

create the new polynomial $\displaystyle Q(x) = P(x-k),$ then $\displaystyle Q(x) = b_nx^n + b_{n-2}x^{n-2} + \cdots + b_1x + b_0$ meaning that the shift has caused the new polynomial to lose its $\displaystyle x^{n-1}$ term.

I know it has something to do with the significance of the 2nd coefficient polynomial, but I get lost after that. any help appreciated.
Hint: $\displaystyle k=\frac{a_{n-1}}{na_n}.$