Polynomial Division and Remainder Theorem

Why Divide Polynomials?

Polynomial division helps us:

• Factor polynomials
• Find zeros of polynomials
• Simplify rational expressions
• Analyze polynomial behavior

Long Division of Polynomials

The process is similar to long division with numbers.

Steps for Polynomial Long Division

1. Arrange both polynomials in descending order of powers
2. Divide the leading term of the dividend by the leading term of the divisor
3. Multiply the result by the divisor
4. Subtract from the dividend
5. Bring down the next term
6. Repeat until the degree of the remainder is less than the degree of the divisor

Example Format

$\frac{\text{dividend}}{\text{divisor}} = \text{quotient} + \frac{\text{remainder}}{\text{divisor}}$

Synthetic Division

Synthetic division is a shortcut for dividing by linear factors of the form $(x - c)$.

When to Use Synthetic Division

✓ Divisor is $(x - c)$ (linear with leading coefficient 1) ✗ Cannot use for divisors like $(2x - 3)$ or $(x^2 + 1)$

Steps for Synthetic Division

1. Write $c$ (from $x - c$) outside the box
2. Write coefficients of the dividend inside
3. Bring down the first coefficient
4. Multiply by $c$, add to next coefficient
5. Repeat across all coefficients
6. Last number is the remainder

Example Setup

Dividing $f(x)$ by $(x - c)$:

c & a_n & a_{n-1} & \cdots & a_0 \\ & & ca_n & \cdots & \\ \hline & a_n & b_{n-1} & \cdots & r \end{array}$$ The bottom row gives quotient coefficients and remainder $r$. ## The Remainder Theorem **Remainder Theorem**: When a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$. $$f(x) = (x - c) \cdot q(x) + r$$ where $r = f(c)$ ### Why It Matters - Quick way to find remainders without full division - Just evaluate $f(c)$! ## The Factor Theorem **Factor Theorem**: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. In other words: - If $f(c) = 0$, then $(x - c)$ divides evenly into $f(x)$ - If the remainder is 0, then $c$ is a zero of $f(x)$ ### Applications 1. **Testing for factors**: Check if $f(c) = 0$ 2. **Finding zeros**: If $(x - c)$ is a factor, then $c$ is a zero 3. **Factoring**: Use known zeros to write factored form ## Rational Zero Theorem If $f(x) = a_nx^n + ... + a_0$ has integer coefficients, then any rational zero $\frac{p}{q}$ must have: - $p$ divides the constant term $a_0$ - $q$ divides the leading coefficient $a_n$ This gives us a list of **possible** rational zeros to test. ## Complete Factorization Strategy 1. Use Rational Zero Theorem to list possible zeros 2. Test candidates using synthetic division or direct evaluation 3. Once you find a zero $c$, factor out $(x - c)$ 4. Repeat on the quotient polynomial 5. Factor completely over the real numbers