Algebra: Polynomial Definition and Characteristics

Polynomial Definition

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• The variables are raised to non-negative integer powers.

Polynomial Characteristics

• Degree: The highest power of the variable in the polynomial.
• Terms: Individual parts of the polynomial, separated by + or - signs.
• Coefficients: Numbers that multiply the variables in each term.
• Monomials: Polynomials with only one term.
• Binomials: Polynomials with two terms.
• Trinomials: Polynomials with three terms.

Polynomial Operations

• Addition and Subtraction: Combine like terms.
• Multiplication: Distribute each term in one polynomial to each term in the other polynomial.
• Division: Divide each term in the dividend by the divisor, and then simplify.

Polynomial Types

• Monic: A polynomial with a leading coefficient of 1.
• Constant: A polynomial with a degree of 0.
• Linear: A polynomial with a degree of 1.
• Quadratic: A polynomial with a degree of 2.
• Cubic: A polynomial with a degree of 3.

Polynomial Theorems

• Remainder Theorem: The remainder of a polynomial divided by (x - a) is equal to the value of the polynomial at x = a.
• Factor Theorem: If (x - a) is a factor of a polynomial, then the polynomial has a root at x = a.
• Fundamental Theorem of Algebra: Every non-constant polynomial has at least one complex root.

Polynomial Definition and Characteristics

• A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Variables are raised to non-negative integer powers.
• The degree of a polynomial is the highest power of the variable.
• Terms are individual parts of the polynomial, separated by + or - signs.
• Coefficients are numbers that multiply the variables in each term.
• Monomials are polynomials with only one term.
• Binomials are polynomials with two terms.
• Trinomials are polynomials with three terms.

Polynomial Operations

• Adding and subtracting polynomials involves combining like terms.
• Multiplying polynomials involves distributing each term in one polynomial to each term in the other polynomial.
• Dividing polynomials involves dividing each term in the dividend by the divisor, and then simplifying.

Polynomial Types

• Monic polynomials have a leading coefficient of 1.
• Constant polynomials have a degree of 0.
• Linear polynomials have a degree of 1.
• Quadratic polynomials have a degree of 2.
• Cubic polynomials have a degree of 3.

Polynomial Theorems

• The Remainder Theorem states that the remainder of a polynomial divided by (x - a) is equal to the value of the polynomial at x = a.
• The Factor Theorem states that if (x - a) is a factor of a polynomial, then the polynomial has a root at x = a.
• The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex root.

Polynomial Definition

• A polynomial consists of variables and coefficients combined using only addition, subtraction, and multiplication.
• Variables are raised to non-negative integer powers.

Types of Polynomials

• A monomial is a polynomial with only one term, e.g. 3x^2 or 5y.
• A binomial is a polynomial with two terms, e.g. x^2 + 3x or y^2 - 4y.
• A trinomial is a polynomial with three terms, e.g. x^2 + 2x + 1 or y^3 - 2y^2 + y.

Properties of Polynomials

• The commutative property states that the order of terms does not change the polynomial.
• The associative property states that the order in which terms are grouped does not change the polynomial.
• The distributive property allows a single value or expression to be distributed to each term inside parentheses.

Operations with Polynomials

• To add or subtract polynomials, combine like terms.
• To multiply polynomials, multiply each term in one polynomial by each term in the other.

Degree of a Polynomial

• The degree of a polynomial is the highest power of the variable(s).
• For example, the degree of x^2 + 3x - 1 is 2.

Zeros of a Polynomial

• A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
• For example, the zeros of x^2 + 3x + 2 are x = -1 and x = -2.