Polynomial Operations

A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Example: 3x^2 + 2x - 5.

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 add polynomials, combine like terms (terms with the same variable and exponent). Example: (3y^2 + y^3 - 5) + (4y^2 - 4y + 2y^3 + 8) = 3y^3 + 7y^2 - 4y + 3.

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

To subtract polynomials, distribute the negative sign to the second polynomial and then combine like terms. Example: (5x + x^2 - 4) - (4x - x^2 + 6) = 2x^2 + x - 10.

The product of a binomial squared (a + b)^2 is calculated using the formula: (a + b)^2 = a^2 + 2ab + b^2. Example: (2x + 3)^2 = 4x^2 + 12x + 9.

The distributive property states that a(b + c) = ab + ac. It is used to multiply a single term by a polynomial. Example: -3x^2(7x^2 - x + 4) = -21x^4 + 3x^3 - 12x^2.

To simplify a polynomial expression, combine like terms and apply the distributive property where necessary. Example: Simplifying (x - 3)^2 gives x^2 - 6x + 9.

A coefficient is a numerical factor in a term of a polynomial. In the term 5x^2, 5 is the coefficient.

A constant term is a term in a polynomial that does not contain any variables. Example: In the polynomial 3x^2 + 2x - 5, -5 is the constant term.