Polynomial Operations

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 2x^2 + 3x - 5 is a polynomial.

Like terms are terms in a polynomial that have the same variable raised to the same power. For example, in the expression 3x + 5x - 2, the terms 3x and 5x are like terms.

To add polynomials, combine like terms by adding their coefficients. For example, (2x^2 + 3x) + (4x^2 + 5x) = (2x^2 + 4x^2) + (3x + 5x) = 6x^2 + 8x.

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

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

The product of two binomials can be found using the distributive property or the FOIL method (First, Outside, Inside, Last). For example, (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

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.

To multiply a polynomial by a monomial, distribute the monomial to each term in the polynomial. For example, 3x(2x^2 + 4x) = 6x^3 + 12x^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 - 2)(x + 3) = 0, then x - 2 = 0 or x + 3 = 0.

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.