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