Chapter 4 Section 3: Polynomial Division and Synthetic Division

Welcome back to Professor Baker's Math Class! In this section, we'll be exploring the fascinating world of polynomial division. Specifically, we will focus on two methods: long division and synthetic division. These techniques are essential for simplifying rational expressions, factoring polynomials, and solving higher-degree polynomial equations.

Why Polynomial Division?

Polynomial division helps us rewrite complex polynomial fractions into simpler forms, similar to how we simplify regular fractions. This simplification is crucial for various applications in calculus and other advanced math courses.

Long Division of Polynomials

Long division with polynomials might seem intimidating at first, but it follows the same logic as long division with numbers. Here’s a quick recap of the key steps:

  1. Arrange: Make sure both the dividend and the divisor are written in descending order of exponents. Include placeholders (terms with a coefficient of 0) for any missing terms.
  2. Divide: Divide the leading term of the dividend by the leading term of the divisor. This gives you the first term of the quotient.
  3. Multiply: Multiply the entire divisor by the term you just obtained in the quotient.
  4. Subtract: Subtract the result from the dividend.
  5. Bring Down: Bring down the next term from the dividend.
  6. Repeat: Repeat steps 2-5 until there are no more terms to bring down.

The final result will be in the form: $$\frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$$.

For example, let's divide $x^3 + 2x^2 - 6x - 9$ by $x - 2$. After performing long division, we find that the quotient is $x^2 + 4x + 2$ and the remainder is $-5$. Therefore, we can write: $$\frac{x^3 + 2x^2 - 6x - 9}{x - 2} = x^2 + 4x + 2 - \frac{5}{x - 2}$$.

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear divisor of the form $x - c$. It's much faster and more efficient than long division when applicable.

Here's how synthetic division works:

  1. Write Down: Write down the coefficients of the dividend. Remember to include zeros for any missing terms!
  2. Write 'c': Write the value of 'c' (from the divisor $x-c$) to the left.
  3. Bring Down: Bring down the first coefficient.
  4. Multiply: Multiply the value you just brought down by 'c'.
  5. Add: Add the result to the next coefficient.
  6. Repeat: Repeat steps 4 and 5 until you reach the last coefficient.

The last number you obtain is the remainder. The other numbers are the coefficients of the quotient, which will have a degree one less than the original dividend.

Using the same example as before, dividing $x^3 + 2x^2 - 6x - 9$ by $x - 2$ using synthetic division, we get the same quotient ($x^2 + 4x + 2$) and remainder ($-5$).

The Remainder Theorem

The Remainder Theorem states that when you divide a polynomial $f(x)$ by $x - c$, the remainder is equal to $f(c)$. This provides a quick way to evaluate a polynomial at a specific value. In our example, the remainder when dividing by $x-2$ was $-5$. According to the Remainder Theorem, $f(2) = -5$.

The Factor Theorem

The Factor Theorem is a direct consequence of the Remainder Theorem. It states that $x - c$ is a factor of $f(x)$ if and only if $f(c) = 0$. In other words, if the remainder after dividing by $x-c$ is zero, then $x-c$ is a factor of the polynomial.

Understanding polynomial and synthetic division opens doors to more advanced topics in algebra and calculus. Practice these techniques, and you'll find them invaluable in your mathematical journey. Keep practicing, and don't hesitate to ask questions!