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Questions and Answers
What is the general form of a polynomial and what does 'n' represent in the expression?
What is the general form of a polynomial and what does 'n' represent in the expression?
The general form of a polynomial is a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0, where 'n' represents the degree of the polynomial, which is the highest power of the variable.
What is the difference between a monomial and a binomial?
What is the difference between a monomial and a binomial?
A monomial is a polynomial with only one term, while a binomial is a polynomial with two terms.
What is the Fundamental Theorem of Algebra and what does it state about the zeroes of a polynomial?
What is the Fundamental Theorem of Algebra and what does it state about the zeroes of a polynomial?
The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex zero.
What is the relationship between the degree of a polynomial and the number of its zeroes?
What is the relationship between the degree of a polynomial and the number of its zeroes?
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What is the Remainder Theorem and how can it be used?
What is the Remainder Theorem and how can it be used?
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What is the Factor Theorem and what does it state about the relationship between zeroes and factors of a polynomial?
What is the Factor Theorem and what does it state about the relationship between zeroes and factors of a polynomial?
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What is a constant polynomial and how does it differ from a zero polynomial?
What is a constant polynomial and how does it differ from a zero polynomial?
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What is the difference between a trinomial and a binomial?
What is the difference between a trinomial and a binomial?
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What is the purpose of the Fundamental Theorem of Algebra and how does it relate to the number of zeroes of a polynomial?
What is the purpose of the Fundamental Theorem of Algebra and how does it relate to the number of zeroes of a polynomial?
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Study Notes
Definition of Polynomials
- A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- The general form of a polynomial is:
a_n x^n + a_(n-1) x^(n-1) +... + a_1 x + a_0, where:-
a_nis the leading coefficient (non-zero) -
xis the variable -
nis the degree of the polynomial (highest power of the variable)
-
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 -
Constant Polynomial: A polynomial with only a constant term, e.g.,
5 -
Zero Polynomial: A polynomial with all coefficients equal to zero, e.g.,
0x^2 + 0x + 0
Zeroes of Polynomials
- A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
- The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex zero.
- The number of zeroes of a polynomial is equal to its degree, counting multiplicities.
Remainder Theorem
- The Remainder Theorem states that if a polynomial
f(x)is divided byx - a, the remainder isf(a). - This theorem can be used to find the value of a polynomial at a specific point.
Factor Theorem
- The Factor Theorem states that if
ais a zero of a polynomialf(x), thenx - ais a factor off(x). - Conversely, if
x - ais a factor off(x), thenais a zero off(x). - This theorem can be used to find the factors of a polynomial.
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Description
Learn about the definition, types, and important theorems related to polynomials, including the Remainder and Factor Theorems. Test your understanding of polynomial concepts and their applications.
