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.
Dividing a Monomial by a Monomial
Divide coefficients, then use the index law: xⁿ ÷ xᵐ = xⁿ⁻ᵐ
12x³ ÷ 4x = (12/4) × x³⁻¹ = 3x²
−15a²b ÷ 5ab = −3 × a²⁻¹ × b¹⁻¹ = −3a
Dividing a Polynomial by a Monomial
Divide each term of the polynomial separately by the monomial.
(6x² + 9x) ÷ 3x
6x² ÷ 3x = 2x. 9x ÷ 3x = 3. Answer: 2x + 3
(10a²b − 4ab²) ÷ 2ab
10a²b ÷ 2ab = 5a. −4ab² ÷ 2ab = −2b. Answer: 5a − 2b
Introduction to Polynomial Long Division
(x² + 5x + 6) ÷ (x + 2)
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.
Practice Questions
- Simplify 20x² ÷ 5x.
- Simplify (8x³ + 12x²) ÷ 4x.
- Simplify −18a²b ÷ 6ab.
- Divide (x² + 7x + 10) by (x + 2).
- Simplify (15m²n − 9mn²) ÷ 3mn.