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Polynomial division is a method similar to long division but specifically used for dividing polynomials. It involves dividing a polynomial (the dividend) by another polynomial (the divisor) to get a quotient and possibly a remainder.

When dividing polynomials, you may encounter terms like coefficients, which are the numbers in front of the variables, and exponents, which show the power to which the variables are raised.

For example, in the polynomial \(a^3 - 10a + 12\), the coefficients are 1, 0, -10, and 12, while the exponents are 3, 1, and 0. Polynomial division can be performed manually, much like basic arithmetic division, or using specific techniques like synthetic division.

Synthetic division is a simplified method for dividing polynomials, particularly when dividing by a linear factor like \(a - 2\). This method saves time and reduces complexity.

Here are the steps for synthetic division:

• Write down the coefficients of the polynomial. For \(a^3 - 10a + 12\), the coefficients are \{1, 0, -10, 12\}.
• Determine the zero of the divisor. For \(a - 2\), the zero is 2.
• Set up the synthetic division by placing the zero to the left and the coefficients to the right:

\(\begin{array}{r|rrrr}2 & 1 & 0 & -10 & 12 \ \ \end{array}\).

• Bring down the first coefficient.
• Multiply the zero of the divisor by the number you've just written down.
• Add this result to the next coefficient and write it below.
• Repeat the process until all coefficients have been used.

You finally get the bottom row which contains the coefficients of the quotient and possibly the remainder.

Let's return to our example \(a^3 - 10a + 12\) divided by \(a - 2\) to illustrate synthetic division in practice:

• List coefficients: \{1, 0, -10, 12\}
• Zero of the divisor: 2
• Setup: \(\begin{array}{r|rrrr}2 & 1 & 0 & -10 & 12 \ \ \end{array}\)
• Bring down the first coefficient (1).

Now perform the operations:

\(\begin{array}{r|rrrr}2 & 1 & 0 & -10 & 12 \ \ & & 2 & 4 & -12 \hline \ & 1 & 2 & -6 & 0 \end{array}\).- First, bring down 1.
- Multiply: 2 * 1 = 2. Add to 0: 0 + 2 = 2.
- Multiply: 2 * 2 = 4. Add to -10: -10 + 4 = -6.
- Multiply: 2 * -6 = -12. Add to 12: 12 - 12 = 0.
The bottom row \{1, 2, -6, 0\} represents the quotient and the remainder. The remainder is zero, so the quotient is \(a^2 + 2a - 6\). Synthetic division provides a compact and efficient way to divide polynomials.