Welcome back to class! Today, we explored a crucial tool in our Pre-Calculus toolkit: Polynomial Long Division. While factoring quadratics is often straightforward, handling polynomials of degree 3 or higher requires a more robust approach. Long division allows us to break these complex expressions down to find their roots.

Why Do We Need Long Division?

When we cannot easily factor by grouping, long division helps us test potential factors. If we divide a polynomial $P(x)$ by a divisor $D(x)$ and find that the remainder is zero, we know that $D(x)$ is a factor of the polynomial. This is the essence of the Factor Theorem.

The Division Algorithm

Recall that when we perform division, we are reorganizing the expression into the following form:

$$P(x) = D(x) \cdot Q(x) + R(x)$$

Where:

  • $P(x)$ is the Dividend (what we are dividing).
  • $D(x)$ is the Divisor (what we are dividing by).
  • $Q(x)$ is the Quotient (our answer).
  • $R(x)$ is the Remainder.

Step-by-Step Strategy: DMSB

The process mirrors long division with numbers. We follow the DMSB cycle:

  1. Divide: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
  2. Multiply: Multiply that new term in the quotient by the entire divisor.
  3. Subtract: Subtract this result from the dividend. (Pro Tip: Be very careful with your negative signs here! It is often easier to flip the signs and add).
  4. Bring Down: Bring down the next term from the dividend.

Repeat this cycle until the degree of the remainder is less than the degree of the divisor.

Crucial Tip: The Zero Placeholder

One of the most common mistakes happens when terms are "missing" in the polynomial. For example, if you are dividing $x^3 - 8$, you are skipping the $x^2$ and $x$ terms.

Always rewrite the polynomial in standard form with placeholders using a zero coefficient:

$$x^3 + 0x^2 + 0x - 8$$

This ensures your columns align correctly during the subtraction phase, preventing simple arithmetic errors.

Conclusion

Long division is the stepping stone to Synthetic Division and finding all zeros of a function. If your remainder is $0$, congratulations! You have successfully factored the polynomial. If not, the remainder can still tell us the value of the function at that point (the Remainder Theorem).

Be sure to review the Class Notes from 10-11-08 attached below for detailed examples and practice problems.