Polynomial Division
What is a Polynomial?
A polynomial is an algebraic expression made up of a finite number of terms, where every power of the variable (usually x) is a nonnegative integer. Think of it as a sum of terms like 3x + 5, or 2x² − 4y + 6, or x³ + 6x² − 9x − 14.
What counts as a polynomial?
These ARE polynomials: 3x + 5, 2x² − 4y + 6, 8 (a plain number is a polynomial). These are NOT: √x (fractional index), 5x⁻³ (negative index), 3/x (negative index), x = 3x⁻¹ (negative index).
What is Polynomial Division?
Polynomial division is a method for splitting a polynomial into a product of smaller polynomials — with or without a remainder left over. It's the exact same process as the long division ("bus stop division") you learned for dividing regular numbers.
At A level, you'll typically divide a cubic (degree 3) or quartic (degree 4) polynomial by a linear divisor in the form (x ± p). The answer you get is built up term by term, working downwards in powers of x.
How to Divide Polynomials: Step by Step
The method is called long division, and it works exactly like dividing numbers. Here's the process:
- Start with the highest power term of the dividend. Divide it by the highest power term of the divisor. This is your first quotient term.
- Multiply this quotient term by the entire divisor, and write the result below.
- Subtract this result from the dividend.
- Bring down the next term, and repeat with this new remainder.
- Keep going until you reach the end. If you're left with zero, the divisor is a factor. If not, what's left is your remainder.
Worked Example: Divide f(x) = x³ + 6x² − 9x − 14 by (x − 2)
Step 1: Divide x³ by x → get x²
x² × (x − 2) = x³ − 2x²
Subtract: (x³ + 6x²) − (x³ − 2x²) = 8x²
Step 2: Bring down −9x, now we have 8x² − 9x
Divide 8x² by x → get 8x
8x × (x − 2) = 8x² − 16x
Subtract: (8x² − 9x) − (8x² − 16x) = 7x
Step 3: Bring down −14, now we have 7x − 14
Divide 7x by x → get 7
7 × (x − 2) = 7x − 14
Subtract: (7x − 14) − (7x − 14) = 0
So: x³ + 6x² − 9x − 14 = (x − 2)(x² + 8x + 7) with NO remainder
Remainder vs. No Remainder: If the divisor divides evenly (remainder = 0), then the divisor is a factor of the polynomial. If there's a remainder left, write it as a separate term: f(x) = (x − a) × Q(x) + R, where Q(x) is the quotient and R is the remainder.
Divide f(x) = x³ + 2x² + 3x + 4 by (x − 1). Is there a remainder?