Integrated Math 3 / Unit 1: Polynomial Functions
1.4 Polynomial Division
Notes
You can divide polynomials like numbers: $\displaystyle \frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}$.
- Long division: divide leading terms, multiply, subtract, bring down, repeat. Include a zero coefficient for any missing power.
- Synthetic division works for divisors $x - c$. Use $c$ (so for $x + 4$ use $-4$).
- Example: divide $x^3 + 2x^2 - 5x - 6$ by $x - 2$. Using $c = 2$ on coefficients $1, 2, -5, -6$ gives $1, 4, 3$ and remainder $0$. The quotient is $x^2 + 4x + 3$.
- A remainder of 0 means the divisor is a factor.
Practice
Divide $2x^3 - 3x^2 + x + 5$ by $x + 1$ using synthetic division. What is the remainder?
Use $c = -1$ on $2, -3, 1, 5$: bring down 2; $-3 - 2 = -5$; $1 + 5 = 6$; $5 - 6 = -1$. The quotient is $2x^2 - 5x + 6$.
What number do you use in synthetic division by $x + 4$?
Write $x + 4 = x - (-4)$, so $c = -4$.
Divide $x^2 + 5x + 7$ by $x + 2$. The result is:
$(x + 2)(x + 3) = x^2 + 5x + 6$, which leaves a remainder of $7 - 6 = 1$.
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