Polynomial Operations

A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. For example, 4x^3 + 3x^2 - 2x + 1 is a polynomial.

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial 5x^4 + 3x^2 - 2, the degree is 4.

To add polynomials, combine like terms by adding their coefficients. For example, (2x^2 + 3x) + (4x^2 + 5) = (2x^2 + 4x^2) + 3x + 5 = 6x^2 + 3x + 5.

Like terms are terms that have the same variable raised to the same power. For example, 3x^2 and 5x^2 are like terms, while 3x^2 and 4x are not.

To subtract polynomials, distribute the negative sign and then combine like terms. For example, (5x^2 + 3) - (2x^2 + 1) = (5x^2 - 2x^2) + (3 - 1) = 3x^2 + 2.

The standard form of a polynomial is when the terms are arranged in descending order of their degrees. For example, 4x^3 + 2x^2 - x + 5 is in standard form.

A monomial is a polynomial with only one term, such as 3x^2. A polynomial can have multiple terms, such as 2x^2 + 3x + 1.

The coefficient is the numerical factor in front of a variable in a term. For example, in 7x^3, 7 is the coefficient.

To multiply polynomials, use the distributive property (FOIL for binomials) to multiply each term in one polynomial by each term in the other. For example, (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

The result is 11x^4 + 2x^2 + 14x.