Use synthetic division to find the quotient and remainder when $9 x^{6}-5 x^{4}+5 x^{2}+6$ is divided by $x-2$.
The quotient is The remainder is
Answer
The quotient is $9x^5+18x^4+31x^3+62x^2+129x+258$ and the remainder is $\boxed{522}$.
Step 1 ：First, we set up the synthetic division with $2$ as the divisor and the coefficients of $9 x^{6}-5 x^{4}+5 x^{2}+6$ as the dividend.
Step 2 ：\[\begin{array}{rrrrrrrr} \multicolumn{1}{r|}{2} & {9} & 0 & -5 & 0 & 5 & 0 & 6 \\ \multicolumn{1}{r|}{} & & 18 & 36 & 62 & 124 & 258 & 516 \\ \cline{2-8} & 9 & 18 & 31 & 62 & 129 & 258 & \multicolumn{1}{|r}{522} \end{array}\]
Step 3 ：Thus, we find that $9 x^{6}-5 x^{4}+5 x^{2}+6=(x-2)(9x^5+18x^4+31x^3+62x^2+129x+258)+522$.
Step 4 ：The quotient is $9x^5+18x^4+31x^3+62x^2+129x+258$ and the remainder is $\boxed{522}$.
