Polynomial Division

Why divide polynomials?

Division shows up when you want to:

• Simplify a rational expression.
• Pull a known factor out of a polynomial.
• Evaluate $p(c)$ using the remainder theorem — sometimes faster than plugging in.

Two methods: long division (always works) and synthetic division (faster, but only for divisors of the form $x - c$).

Polynomial long division

Just like numerical long division.

Example. Divide $x^3 + 2x^2 - 5x - 6$ by $x - 2$.

1. $x^3 \div x = x^2$. Multiply back: $x^2(x - 2) = x^3 - 2x^2$. Subtract: $(x^3 + 2x^2) - (x^3 - 2x^2) = 4x^2$.
2. Bring down the next term: $4x^2 - 5x$. Divide: $4x^2 \div x = 4x$. Multiply: $4x(x - 2) = 4x^2 - 8x$. Subtract: $4x^2 - 5x - (4x^2 - 8x) = 3x$.
3. Bring down: $3x - 6$. Divide: $3x \div x = 3$. Multiply: $3(x - 2) = 3x - 6$. Subtract: $0$.

Quotient: $x^2 + 4x + 3$. Remainder: $0$.

So $x^3 + 2x^2 - 5x - 6 = (x - 2)(x^2 + 4x + 3)$.

Synthetic division

A shortcut when the divisor is $x - c$. Write only the coefficients.

Example. Same problem, $x^3 + 2x^2 - 5x - 6$ divided by $x - 2$. Use $c = 2$.

2 | 1   2  -5  -6
  |     2   8   6
  |________________
    1   4   3   0

Process: bring down $1$. Multiply by $2 \to 2$. Add: $2 + 2 = 4$. Multiply by $2 \to 8$. Add: $-5 + 8 = 3$. Multiply by $2 \to 6$. Add: $-6 + 6 = 0$.

Bottom row reads off the quotient $x^2 + 4x + 3$ and remainder $0$.

The remainder theorem

If you divide a polynomial $p(x)$ by $x - c$, the remainder equals $p(c)$.

Example. Find $p(3)$ for $p(x) = 2x^3 - x^2 + 4$. Use synthetic division with $c = 3$, coefficients $2, -1, 0, 4$:

3 | 2  -1   0   4
  |    6  15  45
  |_______________
    2   5  15  49

Remainder $= 49 = p(3)$.

A surprisingly fast way to evaluate.

The factor theorem

A corollary: $x - c$ is a factor of $p(x)$ if and only if $p(c) = 0$. Roots and linear factors are two sides of the same coin.

Key takeaways

• Long division works for any polynomial divisor.
• Synthetic division is faster, but only for divisors of the form $x - c$.
• Remainder theorem: dividing $p(x)$ by $x - c$ gives a remainder of $p(c)$.
• Factor theorem: $p(c) = 0 \iff (x - c)$ is a factor.