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Questions and Answers
What is the degree of a polynomial?
What is the degree of a polynomial?
How do you add or subtract polynomials?
How do you add or subtract polynomials?
What is a monomial?
What is a monomial?
What is the Remainder Theorem?
What is the Remainder Theorem?
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What is a quadratic polynomial?
What is a quadratic polynomial?
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How do you multiply polynomials?
How do you multiply polynomials?
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What is a polynomial with a leading coefficient of 1?
What is a polynomial with a leading coefficient of 1?
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What is the Fundamental Theorem of Algebra?
What is the Fundamental Theorem of Algebra?
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What is a polynomial with three terms?
What is a polynomial with three terms?
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What is the difference between a binomial and a trinomial?
What is the difference between a binomial and a trinomial?
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What property of polynomials allows you to reorder the terms in an expression without changing its value?
What property of polynomials allows you to reorder the terms in an expression without changing its value?
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What is the result of distributing 2 to the terms inside the parentheses in the expression 2(x^2 + 3x)?
What is the result of distributing 2 to the terms inside the parentheses in the expression 2(x^2 + 3x)?
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What is the highest power of the variable in the polynomial x^3 - 2x^2 + x?
What is the highest power of the variable in the polynomial x^3 - 2x^2 + x?
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What is a value of the variable that makes the polynomial x^2 + 2x - 3 equal to zero?
What is a value of the variable that makes the polynomial x^2 + 2x - 3 equal to zero?
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What type of polynomial has only one term, such as 4x^2?
What type of polynomial has only one term, such as 4x^2?
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What is the result of adding the polynomials x^2 + 2x and x^2 - 3x?
What is the result of adding the polynomials x^2 + 2x and x^2 - 3x?
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What property of polynomials allows you to regroup terms in an expression without changing its value?
What property of polynomials allows you to regroup terms in an expression without changing its value?
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What is the result of multiplying the polynomials x^2 + 2x and x + 1?
What is the result of multiplying the polynomials x^2 + 2x and x + 1?
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Study Notes
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.
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Description
Learn about the definition and characteristics of polynomials, including degree, terms, coefficients, and monomials.
