use synthetic division to find the remainder. \n$\frac{x^{5}-10x^{3}-x^{2}-6}{x-3}$

Explanation:

Step1: Identify root and coefficients

Root of divisor $x-3$ is $3$. Coefficients of dividend: $1, 0, -10, -1, 0, -6$ (for $x^5, x^4, x^3, x^2, x, \text{constant}$)

Step2: Set up synthetic division

Bring down the leading coefficient: $1$ Multiply by 3: $1\times3=3$ Add to next coefficient: $0+3=3$

Step3: Continue synthetic division

Multiply $3$ by 3: $3\times3=9$ Add to next coefficient: $-10+9=-1$

Step4: Next synthetic division step

Multiply $-1$ by 3: $-1\times3=-3$ Add to next coefficient: $-1+(-3)=-4$

Step5: Next synthetic division step

Multiply $-4$ by 3: $-4\times3=-12$ Add to next coefficient: $0+(-12)=-12$

Step6: Final synthetic division step

Multiply $-12$ by 3: $-12\times3=-36$ Add to last coefficient: $-6+(-36)=-42$

Answer:

$-42$