Algebraic Expression Simplifier
Combine Like Terms · Expand · Multiply · Factor
Simplify polynomial expressions step by step. Supports combining like terms, distributing brackets, FOIL multiplication, and quadratic factoring with multi-variable polynomials.
Quick Examples
Use ^ for powers (x^2), * for multiplication (2*x or just 2x). Variables: x, y, z.
Step-by-Step Solution
Quick Examples
Step-by-Step Solution
Quick Examples
Distribution Steps
Quick Examples
Enter a quadratic expression in x. Constants a, b, c can be any real numbers.
Step-by-Step Solution
What Is Algebraic Expression Simplification?
Simplifying an algebraic expression means rewriting it in its most compact equivalent form without changing its value. The main techniques are combining like terms, expanding parentheses using the distributive property, multiplying polynomials, and factoring to reverse expansion. Mastering these skills is the foundation of all algebra.
Combining Like Terms
Like terms are monomials that share exactly the same variable factors at the same powers. For example, 5x² and -3x² are like terms because both are multiples of x², so their coefficients can be added: 5x² + (-3x²) = 2x². Unlike terms — such as 4x and 4x² — cannot be combined because their variable parts differ.
The step-by-step process is: (1) list every term, (2) group terms by their variable signature, (3) sum the coefficients within each group, and (4) write the result in descending degree order.
Example
3x² + 2x − x² + 5x − 3
Group x² terms: 3x² − x² = 2x²
Group x terms: 2x + 5x = 7x
Constant: −3
Result: 2x² + 7x − 3
The Distributive Property and Expanding Brackets
The distributive property states a(b + c) = ab + ac. When two polynomial factors are adjacent — like (2x + 3)(x − 4) — every term of the first factor must be multiplied by every term of the second. For two binomials, this is commonly described by the FOIL method: First, Outer, Inner, Last.
FOIL Example: (2x+3)(x−4)
| Pair | Product |
|---|---|
| First: 2x × x | 2x² |
| Outer: 2x × (−4) | −8x |
| Inner: 3 × x | 3x |
| Last: 3 × (−4) | −12 |
Combine like terms: 2x² + (−8x + 3x) − 12 = 2x² − 5x − 12
Polynomial Degree and Standard Form
The degree of a monomial is the sum of its variable exponents: 3x²y has degree 3. The degree of a polynomial is the degree of its highest-degree term. Standard form lists terms in descending degree order: 4x³ − 2x² + x − 7. This makes it easy to identify the leading coefficient (4), degree (3), and constant term (−7).
Factoring Quadratics
Factoring reverses expansion. For a quadratic ax² + bx + c, the discriminant D = b² − 4ac determines the nature of roots:
- D > 0: two distinct real roots r₁, r₂ → a(x − r₁)(x − r₂)
- D = 0: one repeated real root r → a(x − r)²
- D < 0: no real factors (roots are complex conjugates)
For example, x² + 5x + 6: D = 25 − 24 = 1, roots = (−5 ± 1)/2 = −2 and −3, so the factored form is (x + 2)(x + 3).
Real-World Applications
- Physics: Kinematic equations use polynomial expressions for position, velocity, and acceleration.
- Engineering: Beam deflection, stress formulas, and control systems rely on polynomial algebra.
- Finance: Compound interest and present value formulas expand into polynomial series.
- Computer Graphics: Bezier curves and surface patches are defined by polynomial equations.
- Data Science: Polynomial regression models fit data using multi-degree polynomials.