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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?
What is the Remainder Theorem and how can it be used?
What is the Remainder Theorem and how can it be used?
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?
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?
What is the difference between a trinomial and a binomial?
What is the difference between a trinomial and a binomial?
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 variablenis 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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