Questions and Answers
The polynomial x^2 + 2x + 1 is a monomial.
False
What is the degree of the polynomial 2x^3 - 4x^2 + x - 1?
What is the result of combining like terms in the polynomial 2x^2 + 3x^2 - 4x + x?
5x^2 - 3x
The _______________________ Theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a)
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Match the following polynomials with their types:
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The polynomial x^2 - 4x + 3 can be factorized into (x - 1)(x - 3).
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What is the definition of a polynomial?
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The polynomial 2x^2 + 3x - 4 has a degree of 3.
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The polynomial x^2 + 2x + 1 can be simplified by combining _______________________ terms.
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What is the term for individual terms in a polynomial?
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Match the following polynomial operations with their descriptions:
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Study Notes
Polynomials
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.
Examples
- 3x^2 + 2x - 4 is a polynomial
- x^2 - 4x + 3 is a polynomial
- 2y^3 - 5y^2 + y - 1 is a polynomial
Properties
- The degree of a polynomial is the highest power of the variable(s) in the expression.
- The leading coefficient is the coefficient of the term with the highest degree.
- Like terms are terms with the same variable(s) raised to the same power.
Operations
- Adding and subtracting polynomials:
- Combine like terms
- Simplify the resulting expression
- Multiplying polynomials:
- Use the distributive property to multiply each term of one polynomial by each term of the other polynomial
- Combine like terms
- Dividing polynomials:
- Use long division or synthetic division to find the quotient and remainder
Special Types of Polynomials
- Monomial: a polynomial with only one term (e.g. 3x^2)
- Binomial: a polynomial with two terms (e.g. x^2 + 3x)
- Trinomial: a polynomial with three terms (e.g. x^2 + 3x + 2)
Theorems and Formulas
- The Remainder Theorem: if a polynomial f(x) is divided by (x - a), the remainder is f(a)
- The Factor Theorem: if a polynomial f(x) has a root at x = a, then (x - a) is a factor of f(x)
- The Fundamental Theorem of Algebra: every non-constant polynomial has at least one complex root
Polynomials
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.
Examples of Polynomials
- 3x^2 + 2x - 4
- x^2 - 4x + 3
- 2y^3 - 5y^2 + y - 1
Properties of Polynomials
Degree and Leading Coefficient
- The degree of a polynomial is the highest power of the variable(s) in the expression.
- The leading coefficient is the coefficient of the term with the highest degree.
Like Terms
- Like terms are terms with the same variable(s) raised to the same power.
Operations with Polynomials
Adding and Subtracting Polynomials
- Combine like terms.
- Simplify the resulting expression.
Multiplying Polynomials
- Use the distributive property to multiply each term of one polynomial by each term of the other polynomial.
- Combine like terms.
Dividing Polynomials
- Use long division or synthetic division to find the quotient and remainder.
Special Types of Polynomials
Monomials, Binomials, and Trinomials
- Monomial: a polynomial with only one term (e.g. 3x^2).
- Binomial: a polynomial with two terms (e.g. x^2 + 3x).
- Trinomial: a polynomial with three terms (e.g. x^2 + 3x + 2).
Theorems and Formulas
Remainder Theorem
- If a polynomial f(x) is divided by (x - a), the remainder is f(a).
Factor Theorem
- If a polynomial f(x) has a root at x = a, then (x - a) is a factor of f(x).
Fundamental Theorem of Algebra
- Every non-constant polynomial has at least one complex root.
Polynomials
Definition and Examples
- A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication, with variables raised to non-negative integer powers.
- Examples of polynomials include 3x^2 + 2x - 4, x^4 - 2x^2 + 1, and 2y^3 - 5y^2 + y - 1.
Terminology
- A monomial is an individual term in a polynomial, such as 3x^2, 2x, or -4.
- The degree of a polynomial is the highest power of the variable, such as 2 in 3x^2 + 2x - 4.
- The leading coefficient of a polynomial is the coefficient of the term with the highest degree, such as 3 in 3x^2 + 2x - 4.
Operations
- When adding or subtracting polynomials, combine like terms, such as (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x.
- When multiplying polynomials, multiply each term in one polynomial by each term in the other, such as (2x + 1)(x + 3) = 2x^2 + 7x + 3.
Properties
- The commutative property states that the order of terms does not change the polynomial, such as 2x^2 + 3x - 4 = 3x - 4 + 2x^2.
- The associative property states that the order of operations does not change the polynomial, such as (2x + 1)(x + 3) = 2x(x + 3) + 1(x + 3).
- The distributive property states that a single value or expression can be distributed to multiple terms, such as 2(x^2 + 3x) = 2x^2 + 6x.
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Description
Learn about polynomials, including their definition, examples, and properties such as degree and leading coefficients. Test your understanding with this quiz!
