Polynomials are algebraic expressions with one or more terms. A standard-form polynomial is written in descending order by degrees of the variables in the terms and set equal to zero. For example, the polynomial 4x^3 + 3x^2 + 5x + 7 = 0 is in standard form, but 2x + 4x^2 = 5 is not. Solving a polynomial in standard form involves chipping away one root at a time, if possible, and using numerical and other complex methods.
Expand the parenthesis terms. For example, expand the polynomial 2x(2 + 4x) to 4x + 8x^2.
Bring the terms over to the left-hand side of the equation and set it to zero. Remember that terms change signs when they are brought from one side of the "=" sign to the other. For example, to solve the polynomial equation, 2x^3 + x^2 + x = 5 - x^3, bring the terms to the left and set the equation to zero: 2x^3 + x^2 + x + x^3 - 5 = 0.
Place like terms next to one another. Like terms are those with the same variable exponents. In the example, the like terms are "2x^3" and "x^3." So the rearranged equation becomes 2x^3 + x^3 + x^2 + x - 5 = 0.
Simplify the expression by adding like terms and factoring out common terms. In the example, 2x^3 + x^3 + x^2 + x - 5 = 0 simplifies to 3x^3 + x^2 + x - 5 = 0. Consider a different polynomial 4x^2 + 8x + 12 = 0. Because 4 is a common factor, the equation can be written as 4(x^2 + 2x + 3) = 0. Dividing both sides by 4, you get x^2 + 2x + 3 = 0. Note that a common factor which includes a variable cannot be discarded because it is one of the solutions to the equation. For example, one of the solutions to the equation, x(x + 1) = 0, is x = 0.
Determine the number of roots or solutions of the polynomial. This is equal to the degree of the polynomial. For example, the polynomial 2x^2 + 3x + 5 = 0 is of degree two and thus has two roots. Some or all of the roots might be the same: For example, the polynomial (x - 2)^2 = 0 has two roots, but they are both x = 2.
Solve the polynomial equation by whittling it down and factoring where possible. Some polynomials can be factored by inspection, while others require more work. Solving a linear equation of the form ax + b = 0, where "a" and "b" are constants, is simple: ax = -b, or x = -b/a. Even quadratic equations of the form ax^2 + bx + c = 0 can sometimes be solved by inspection: For example, the equation x^2 + 3x + 2 = 0 has two factors, x + 1 and x + 2, and so the solutions are x = -1 and x = -2.
