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Unit 3 Polynomial Operations Notes
Terms: The ____________________________ combined by addition or subtraction. Expressions can be: a single
number, variables, and/or variables multiplied by numbers
Degree: A degree is the power of a term.
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The degree is determined by the sum of all the exponents of the variables.
3𝑥
5𝑥𝑦
𝑥3
8𝑥 4 𝑦 3
10𝑤𝑥 2 𝑦 4 𝑧
9
Polynomial: An expression of a sum of terms where an + or – sign separates the terms. Thus, polynomial mean
“______________”. Each polynomial has a term number and degree.
Standard Form: This occurs when the polynomial’s terms are written from ______________ to the ____________.
If more than one term has the ________________, but cannot be combined (different variables), then write
in _____________________.
Polynomial
Polynomial in Standard Form
3+𝑥
−5𝑦 + 6𝑥𝑦 − 3𝑥
14𝑥 3 𝑦 2 − 3𝑥 5 𝑦 2
Coefficient: The number in front of a term.
3𝑥𝑦
The coefficient is
Leading Coefficient: The coefficient of the highest degree term in a polynomial.
−9𝑥 3 + 2𝑥 + 5
The leading coefficient is
Adding and Subtracting Polynomials: When combining like terms, each term must have the exact same
_______________ and _______________ on each variable.
Note: when combining like terms, add/sub the _________________ and keep the _____________ the same.
5𝑥 + 7𝑥
8𝑗 2 − 7𝑗 3
4𝑝3 𝑞 5 − 11𝑞 5 𝑝3
6𝑎7 𝑏 3 + 14𝑎3 𝑏 7
6𝑢𝑣 7 + 12𝑣 7
Multiplying Polynomials: You must ______________ when multiplying. Any term can be multiplied to another.
Thus, ______________ the coefficients and _______ the exponents of same variables. If there are different
variable, leave them separate with their exponents.
(7𝑥 2 )(13𝑥 4 )
(3𝑥𝑦 2 )(−6𝑥 8 𝑧)
(9𝑥 3 )(2𝑥 2 + 7)
(4𝑥 − 2𝑦)(𝑥 2 − 5)
Naming Polynomials: Each polynomial can be classified by the number of terms and overall degree.
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The number of terms is easily identified. Remember, addition and subtraction separate each term.
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The degree of a polynomial is equal to the highest degree term. You do NOT add up all degrees to determine the degree of
the polynomial.
Degree
Naming
Example
0
Constant
4
1
Linear
2𝑥 + 7
2
Quadratic
9𝑦 2 − 𝑥 + 𝑦 + 3
3
Cubic
7𝑥 2 𝑦 + 8𝑥
4
Quartic
𝑥 4 − 2𝑥 3 − 5𝑥 2 + 11𝑥 + 7
5
Quintic
𝑥 2 𝑦𝑧 2
6+
nth Degree
Polynomial
4𝑥 3 𝑦 3 + 3𝑥 2 𝑦 2 − 1
Examples of naming altogether:
9𝑥 − 7𝑥 2 𝑦 + 14𝑦 − 10
13𝑥 5 𝑦𝑧 2
10𝑥 + 14𝑦 − 3𝑧
4𝑧 2 − 25
1,050
Number of Terms
Term Name