Questions: Divide. (3c^3 - 26c - 9) ÷ (c - 3) = with remainder

Solution Steps

To divide the polynomial \(3c^3 - 26c - 9\) by \(c - 3\), we can use polynomial long division or synthetic division. Here, we'll use synthetic division since the divisor is linear. We will set up the synthetic division using the root of the divisor, which is \(c = 3\), and perform the division to find the quotient and remainder.

We are dividing the polynomial \(3c^3 - 26c - 9\) by \(c - 3\). The coefficients of the polynomial are given as \(3, 0, -26, -9\). The root of the divisor \(c - 3\) is \(c = 3\).

Using synthetic division, we start with the leading coefficient \(3\):

• Bring down \(3\).
• Multiply \(3\) by \(3\) (the root) to get \(9\) and add it to the next coefficient \(0\) to get \(9\).
• Multiply \(9\) by \(3\) to get \(27\) and add it to \(-26\) to get \(1\).
• Multiply \(1\) by \(3\) to get \(3\) and add it to \(-9\) to get \(-6\).

The resulting coefficients from the synthetic division are \(3, 9, 1\) with a remainder of \(-6\).

The quotient polynomial is \(3c^2 + 9c + 1\) and the remainder is \(-6\). Therefore, we can express the result of the division as: \[ 3c^3 - 26c - 9 = (c - 3)(3c^2 + 9c + 1) - 6 \]

Final Answer