Algebraic Division – Simplifying Expressions
Dividing algebraic expressions means cancelling common factors. This is the foundation for simplifying fractions, solving equations, and working with rational expressions.
The index law used here (xⁿ ÷ xᵐ = xⁿ⁻ᵐ) is simply the multiplication index law run in reverse, and it is what makes polynomial long division possible — a method structurally identical to the long division of ordinary numbers you learned with whole numbers, just tracking powers of x instead of powers of 10. Polynomial division shows up constantly in more advanced algebra, particularly when factorising a cubic or higher-degree polynomial once one root is already known.
Dividing a Monomial by a Monomial
Divide coefficients, then use the index law: xⁿ ÷ xᵐ = xⁿ⁻ᵐ
Dividing a Polynomial by a Monomial
Divide each term of the polynomial separately by the monomial.
6x² ÷ 3x = 2x. 9x ÷ 3x = 3. Answer: 2x + 3
10a²b ÷ 2ab = 5a. −4ab² ÷ 2ab = −2b. Answer: 5a − 2b
Introduction to Polynomial Long Division
x² ÷ x = x. x(x+2) = x² + 2x. Subtract: 3x + 6. 3x ÷ x = 3. 3(x+2) = 3x + 6. Remainder 0. Answer: x + 3
Key Takeaways
- Divide coefficients normally; subtract exponents for like variables.
- Divide polynomials by monomials term by term.
- Polynomial long division mirrors numerical long division.
- Always check: quotient × divisor = original expression.