Questions and Answers
What is the degree of the polynomial x^3 - 2x^2 + x - 1?
Which of the following is a binomial?
What is the leading coefficient of the polynomial 2x^2 + 3x - 1?
What is the result of adding the polynomials x^2 + 2x and x^2 - 3x?
Signup and view all the answers
What is the result of multiplying the polynomials x + 2 and x + 3?
Signup and view all the answers
What is the factored form of the polynomial 2x^2 + 4x?
Signup and view all the answers
What is the solution to the quadratic equation x^2 + 4x + 4 = 0?
Signup and view all the answers
What is the type of polynomial with three terms?
Signup and view all the answers
What is the constant term in the polynomial x^2 + 3x - 2?
Signup and view all the answers
What is the result of subtracting the polynomial x^2 - 2x from the polynomial x^2 + 3x?
Signup and view all the answers
Study Notes
Polynomial
Definition
- A polynomial is an expression consisting of variables (such as x or y) and coefficients (numbers) combined using only addition, subtraction, and multiplication.
- The variables are raised to non-negative integer powers.
Types of Polynomials
- Monomial: A polynomial with only one term (e.g., 3x^2 or 5y)
- Binomial: A polynomial with two terms (e.g., x^2 + 3x or 2y^2 - 4y)
- Trinomial: A polynomial with three terms (e.g., x^2 + 2x + 1 or y^3 - 2y^2 + y)
Properties of Polynomials
- Degree: The highest power of the variable(s) in a polynomial (e.g., x^2 + 3x has a degree of 2)
- Leading Coefficient: The coefficient of the term with the highest degree (e.g., in x^2 + 3x, the leading coefficient is 1)
- Constant Term: The term with no variables (e.g., in x^2 + 3x + 2, the constant term is 2)
Operations with Polynomials
- Addition and Subtraction: Combine like terms (e.g., (x^2 + 2x) + (x^2 - 3x) = 2x^2 - x)
- Multiplication: Distribute each term in one polynomial to each term in the other polynomial (e.g., (x + 2) × (x + 3) = x^2 + 5x + 6)
Factoring Polynomials
- Factoring out the Greatest Common Factor (GCF): Divide each term by the GCF (e.g., 2x^2 + 4x = 2x(x + 2))
- Factoring Quadratic Expressions: Use the formula x^2 + bx + c = (x + d)(x + e) to factor quadratic expressions
Solving Polynomial Equations
- Linear Equations: Solve for the variable by isolating it on one side of the equation (e.g., 2x + 3 = 5 → x = 1)
- Quadratic Equations: Use factoring or the quadratic formula (x = (-b ± √(b^2 - 4ac)) / 2a) to solve quadratic equations
Polynomial
Definition
- An expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
- Variables are raised to non-negative integer powers.
Types of Polynomials
- Monomial: One-term expression (e.g., 3x^2 or 5y).
- Binomial: Two-term expression (e.g., x^2 + 3x or 2y^2 - 4y).
- Trinomial: Three-term expression (e.g., x^2 + 2x + 1 or y^3 - 2y^2 + y).
Properties of Polynomials
- Degree: Highest power of variables in a polynomial (e.g., x^2 + 3x has a degree of 2).
- Leading Coefficient: Coefficient of the term with the highest degree (e.g., in x^2 + 3x, the leading coefficient is 1).
- Constant Term: Term with no variables (e.g., in x^2 + 3x + 2, the constant term is 2).
Operations with Polynomials
- Addition and Subtraction: Combine like terms (e.g., (x^2 + 2x) + (x^2 - 3x) = 2x^2 - x).
- Multiplication: Distribute each term to each term in the other polynomial (e.g., (x + 2) × (x + 3) = x^2 + 5x + 6).
Factoring Polynomials
- Factoring out the GCF: Divide each term by the Greatest Common Factor (e.g., 2x^2 + 4x = 2x(x + 2)).
- Factoring Quadratic Expressions: Use the formula x^2 + bx + c = (x + d)(x + e) to factor quadratic expressions.
Solving Polynomial Equations
- Linear Equations: Isolate the variable on one side of the equation (e.g., 2x + 3 = 5 → x = 1).
- Quadratic Equations: Use factoring or the quadratic formula (x = (-b ± √(b^2 - 4ac)) / 2a) to solve quadratic equations.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Understand the definition and types of polynomials, including monomials, binomials, and trinomials, and their properties in algebra.
