Questions: Use synthetic division to perform the division. 6a^3 - 53a - 3 / a - 3

Solution Steps

To perform synthetic division, we will use the divisor $a - 3$ to divide the polynomial $6a^3 - 53a - 3$. We will set up the synthetic division by using the root of the divisor, which is $3$, and the coefficients of the polynomial, which are $6, 0, -53, -3$. We will then perform the synthetic division process to find the quotient and remainder.

We are given the polynomial $6a^3 - 53a - 3$ and need to divide it by $a - 3$. The coefficients of the polynomial are $6, 0, -53, -3$, and we will use the root $3$ for synthetic division.

Using synthetic division, we start with the first coefficient $6$:

• Bring down $6$.
• Multiply $6$ by $3$ to get $18$ and add it to the next coefficient $0$ to get $18$.
• Multiply $18$ by $3$ to get $54$ and add it to $-53$ to get $1$.
• Multiply $1$ by $3$ to get $3$ and add it to $-3$ to get $0$.

The results of the synthetic division give us the coefficients of the quotient and the remainder:

• Quotient: $6a^2 + 18a + 1$
• Remainder: $0$

The result of the division can be expressed as: $$\frac{6a^3 - 53a - 3}{a - 3} = 6a^2 + 18a + 1$$

Final Answer