Polynomials in Algebra

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.

Examples

• Simple polynomial: 3x + 2
• Quadratic polynomial: x^2 + 4x + 4
• Cubic polynomial: x^3 - 2x^2 + x - 1

Terminology

• Monomial: a polynomial with one term, e.g. 3x
• Binomial: a polynomial with two terms, e.g. x + 2
• Degree: the highest power of the variable in the polynomial, e.g. the degree of x^2 + 4x + 4 is 2
• Coefficient: a numerical value multiplied by a variable, e.g. the coefficient of x in 3x is 3

Operations

• Addition: polynomials can be added by combining like terms
• Subtraction: polynomials can be subtracted by combining like terms with opposite signs
• Multiplication: polynomials can be multiplied using the distributive property

Properties

• Commutative property: the order of the variables does not change the result
• Associative property: the order in which polynomials are added or multiplied does not change the result
• Distributive property: a polynomial can be multiplied by a sum or difference of polynomials

Types of Polynomials

• Monic polynomial: a polynomial with a leading coefficient of 1
• Constant polynomial: a polynomial with a degree of 0, e.g. 5
• Zero polynomial: a polynomial with all coefficients equal to 0, e.g. 0x + 0

Definition of Polynomials

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

Examples of Polynomials

• Simple polynomial: 3x + 2, consisting of a single variable and a constant.
• Quadratic polynomial: x^2 + 4x + 4, with a degree of 2.
• Cubic polynomial: x^3 - 2x^2 + x - 1, with a degree of 3.

Terminology in Polynomials

• Monomial: a polynomial with one term, e.g. 3x, a single variable with a coefficient.
• Binomial: a polynomial with two terms, e.g. x + 2, consisting of two variables or a variable and a constant.
• Degree: the highest power of the variable in the polynomial, e.g. the degree of x^2 + 4x + 4 is 2.
• Coefficient: a numerical value multiplied by a variable, e.g. the coefficient of x in 3x is 3.

Operations with Polynomials

• Polynomials can be added by combining like terms, e.g. (2x + 3) + (4x + 2) = 6x + 5.
• Polynomials can be subtracted by combining like terms with opposite signs, e.g. (2x + 3) - (4x + 2) = -2x + 1.
• Polynomials can be multiplied using the distributive property, e.g. (2x + 3)(4x + 2) = 8x^2 + 14x + 6.

Properties of Polynomials

• Commutative property: the order of the variables does not change the result, e.g. 2x + 3 = 3 + 2x.
• Associative property: the order in which polynomials are added or multiplied does not change the result, e.g. (2x + 3) + (4x + 2) = 2x + (3 + 4x + 2).
• Distributive property: a polynomial can be multiplied by a sum or difference of polynomials, e.g. 2x(3 + 4x) = 6x + 8x^2.

Types of Polynomials

• Monic polynomial: a polynomial with a leading coefficient of 1, e.g. x^2 + 4x + 4.
• Constant polynomial: a polynomial with a degree of 0, e.g. 5, a single constant value.
• Zero polynomial: a polynomial with all coefficients equal to 0, e.g. 0x + 0, equivalent to zero.