Polynomial Operations

A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. For example, 2x^2 + 3x - 5 is a polynomial.

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

To add polynomials, combine like terms by adding their coefficients. For example, (3x^2 + 2x) + (4x^2 + 5) = 7x^2 + 5.

To subtract polynomials, distribute the negative sign and then combine like terms. For example, (5x^2 - 3) - (2x^2 + 4) = 3x^2 - 7.

The product of two polynomials is found by using the distributive property (FOIL for binomials). For example, (x + 2)(x + 3) = x^2 + 5x + 6.

Like terms are terms that have the same variable raised to the same power. For example, 3x^2 and 5x^2 are like terms.

A monomial is a polynomial with only one term, while a polynomial can have multiple terms. For example, 3x is a monomial, while 3x + 2 is a polynomial.

The standard form of a polynomial is written in descending order of degrees. For example, 4x^3 + 2x^2 - x + 5 is in standard form.

To multiply a polynomial by a monomial, distribute the monomial to each term in the polynomial. For example, 3x(2x^2 + x) = 6x^3 + 3x^2.

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. For example, if (x - 3)(x + 2) = 0, then x = 3 or x = -2.