$f(x) = x^4 - 17x^2 - 36x - 20$\nbinomial: $(x + 2)$\ndivide the given polynomial by the given binomial…

$f(x) = x^4 - 17x^2 - 36x - 20$\nbinomial: $(x + 2)$\ndivide the given polynomial by the given binomial using synthetic division and rewrite the function. if there is no remainder, write 0.\n$g(x) = (x + 2)(\\quad)+(\\quad)$

$f(x) = x^4 - 17x^2 - 36x - 20$\nbinomial: $(x + 2)$\ndivide the given polynomial by the given binomial using synthetic division and rewrite the function. if there is no remainder, write 0.\n$g(x) = (x + 2)(\\quad)+(\\quad)$

Answer

Explanation:

Step1: Identify root for synthetic division

For binomial $(x+2)$, root is $x=-2$. List coefficients of $f(x)=x^4 -17x^2 -36x -20$: $1, 0, -17, -36, -20$ (include 0 for $x^3$ term).

Step2: Set up synthetic division

$$ \begin{array}{r|rrrrr} -2 & 1 & 0 & -17 & -36 & -20 \ \hline & & -2 & 4 & 26 & 20 \ \hline & 1 & -2 & -13 & -10 & 0 \end{array} $$

Step3: Form quotient polynomial

The resulting coefficients $1, -2, -13, -10$ correspond to $x^3 -2x^2 -13x -10$. The final value 0 is the remainder.

Step4: Rewrite the function

Express $f(x)$ as $(x+2)\times$ quotient + remainder.

Answer:

$g(x)=(x + 2)\left(x^3 - 2x^2 - 13x - 10\right)+(0)$