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Definitions, examples, and explanations of monomials and polynomials, including their degrees. Monomials are terms with one variable or constant, while polynomials are sums or differences of monomials. Examples of both and explains how to determine the degree of a polynomial.
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Examples of monomials: 2 𝑥 − 3 𝑥𝑦 1 2
Examples of Polynomials: 2 𝑥 − 4 6 𝑥^2 − 7 𝑥 + 2 𝑥^3 − 2 𝑥^2 + 11 𝑥 − 1 Examples of NON-Polynomials:
1 𝑥
1
The DEGREE of a polynomial is the highest exponent. For example, the following polynomials have the following DEGREES: 7 has a degree of 0 (because 7 ∙ 𝑥^0 = 7 ∙ 1 = 7 ) 2 𝑥 − 4 has a degree of 1 6 𝑥^2 − 7 𝑥 + 2 has a degree of 2 𝑥^3 − 2 𝑥^2 has a degree of 3 𝑥 + 5 𝑥^3 − 2 𝑥^4 − 10 𝑥^2 has a degree of 4 Standard Form of a Polynomial: Polynomials should always be written in standard form. Standard form is when each term is written in descending order of its exponent. For example, 3 𝑥 − 7 𝑥^3 + 9 − 2 𝑥^4 − 𝑥^2 is NOT in standard form. It should be written as: − 2 𝑥^4 − 7 𝑥^3 − 𝑥^2 + 3 𝑥 + 9
Polynomials can be named by the number of terms and/or by the degree.
We just say “a polynomial with degree 4”
call it a polynomial.
Determine if the following expressions are polynomials: 1 4 𝑥^2 + 10 𝑥 yes 4 𝑥 +^10 𝑥^ no^ (x cannot be in the denominator) − 2 𝑥^3 + 𝑥^2 − 7 𝑥 + 1 yes 6 𝑥^3 − 9 𝑥 + 𝑥 1 (^2) + 5 no (the exponent of x can only be 0, 1, 2, 3, 4, …etc) 2 𝑥^3 + 𝑥−^2 − 4 𝑥 + 3 no (the exponent of x can only be 0, 1, 2, 3, 4, …etc) 7 𝑥^ + 1 no (it’s an exponential) 𝑥^7 + 1 yes 𝑥 + 8 1 (^3) yes (it’s the same as x + 2, so it’s linear) |𝑥| + 5 no (it’s an absolute value function) | 5 | + 𝑥 yes (it’s the same as 5 + x, so it’s linear) 4 + 𝑙𝑜𝑔 2 𝑥 no (it’s logarithmic) 𝑥^2 + 3 𝑥 + 𝑙𝑜𝑔 24 yes (it’s the same as 𝑥^2 + 3 𝑥 + 2 , so it’s a quadratic)