Polynomials: Addition, Subtraction, Multiplication

• Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents

Addition and Subtraction of Polynomials

• Polynomial addition involves combining like terms, which are terms with the same variable raised to the same power
• Polynomials can be added by writing them in a vertical format aligning like terms, then adding the coefficients of each column
• Polynomial subtraction can be performed by adding the additive inverse of the polynomial being subtracted
• Distribute the negative sign to each term of the polynomial being subtracted, and then combine like terms as in addition

Multiplication of polynomials

• Polynomial multiplication involves distributing each term of one polynomial to every term of the other polynomial
• Use the distributive property to multiply each term in the first polynomial by each term in the second polynomial
• After distributing, combine like terms to simplify the resulting expression

Division of polynomials

• Polynomial division can be performed using long division or synthetic division
• Long division is similar to numerical long division, arranging the polynomials in descending order of degree and dividing term by term
• Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c
• Synthetic division involves using only the coefficients of the polynomials and the value of c to find the quotient and remainder

Definition of polynomials

• A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables
• A polynomial in one variable (x) can be written in the general form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n, a_{n-1}, ..., a_1, a_0 are constants (coefficients) and n is a non-negative integer (the degree of the polynomial)
• The degree of a polynomial is the highest power of the variable in the polynomial
• The leading coefficient is the coefficient of the term with the highest degree

Properties of polynomials

• Polynomials are closed under addition, subtraction, and multiplication, meaning performing these operations on polynomials always results in another polynomial
• Polynomial division may not always result in a polynomial; it can result in a rational function
• The degree of the sum or difference of two polynomials is less than or equal to the maximum of their individual degrees
• The degree of the product of two polynomials is the sum of their individual degrees

Polynomial equations

• A polynomial equation is an equation in which a polynomial expression is set equal to zero
• The solutions to a polynomial equation are called roots or zeros of the polynomial
• The degree of the polynomial equation determines the maximum number of roots the equation can have

Roots of polynomial equations

• The roots of a polynomial equation are the values of the variable that make the polynomial equal to zero
• Real roots are roots that are real numbers, and they correspond to the x-intercepts of the polynomial function's graph
• Complex roots include imaginary numbers and always occur in conjugate pairs if the coefficients of the polynomial are real

Finding roots

• Factoring is a method to find roots by expressing the polynomial as a product of lower-degree polynomials
• The quadratic formula can be used to find the roots of a quadratic equation (ax^2 + bx + c = 0): x = (-b ± √(b^2 - 4ac)) / (2a)
• The Rational Root Theorem provides a list of possible rational roots of a polynomial equation with integer coefficients
• The Factor Theorem states that if a polynomial f(x) has a root x = c, then (x - c) is a factor of f(x)

Polynomial functions

• A polynomial function is a function defined by a polynomial expression
• The graph of a polynomial function is a smooth, continuous curve
• The end behavior of a polynomial function is determined by its leading term (the term with the highest degree)
• The Intermediate Value Theorem states that if a polynomial function f(x) is continuous on a closed interval [a, b], and f(a) and f(b) have opposite signs, then there exists at least one root of f(x) in the interval (a, b)