Polynomials and Sketching Course notes: 2015 Date edited: 2-July 2015 Page | 1 POLYNOMIALS DEFINITIONS Polynomial: a polynomial, sometimes denoted P(,) is an expression containing only positive whole number powers of Degree: the degree of a polynomial in is the highest power of in the expression Coefficient: the number before a variable in an expression or equation (eg 7 in 7, and -8 in −8 5 ) Monic Polynomial: a polynomial where the leading coefficient of the polynomial is 1. (explained in detail below) Constant term: the term without any x’s.... (eg -4 in 7 − 4) Degree of Polynomial Name 0 1 linear 2 quadratic 3 cubic 4 quartic 5 quintic 6 sextic or hextic 7 septic or heptic 8 octic 9 nonic 10 decic 100 hectic EXAMPLE 1 7 − 4 is a polynomial of degree 1, a linear polynomial 2 + 3 − 10 is a polynomial of degree 2, a quadratic polynomial −2 3 −4 is a polynomial of degree 3, a cubic polynomial 10 is a polynomial of degree 0, (think of it as being 10 0 )
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POLYNOMIALS - Hawker Maths 2018 · 7 −4 is a polynomial of degree 1, a linear polynomial ... Other names of polynomials Degree eight: octic Degree nine: nonic Degree ten: decic
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Polynomials and Sketching Course notes: 2015 Date edited: 2-July 2015 P a g e | 1
POLYNOMIALS
DEFINITIONS
Polynomial: a polynomial, sometimes denoted P(𝑥,) is an expression containing only positive whole number
powers of 𝑥
Degree: the degree of a polynomial in 𝑥 is the highest power of 𝑥 in the expression
Coefficient: the number before a variable in an expression or equation (eg 7 in 7𝑥, and -8 in −8𝑥5)
Monic Polynomial: a polynomial where the leading coefficient of the polynomial is 1. (explained in detail
below)
Constant term: the term without any x’s.... (eg -4 in 7𝑥 − 4)
Degree of Polynomial Name
0
1 linear
2 quadratic
3 cubic
4 quartic
5 quintic
6 sextic or hextic
7 septic or heptic
8 octic
9 nonic
10 decic
100 hectic
EXAMPLE 1
7𝑥 − 4 is a polynomial of degree 1, a linear polynomial
𝑥2 + 3𝑥 − 10 is a polynomial of degree 2, a quadratic polynomial
−2𝑥3 − 4 is a polynomial of degree 3, a cubic polynomial
10 is a polynomial of degree 0, (think of it as being 10𝑥0)
Polynomials and Sketching Course notes: 2015 Date edited: 2-July 2015 P a g e | 2
EXAMPLE 2
The following listed here are NOT polynomials
1
𝑥 𝑥−2 √𝑥 2𝑥 sin 𝑥
Polynomials or not: http://youtu.be/Lc9O-1joI24
CLASSIFICATIONS
Polynomials can also be classified by number of non-zero terms…(how many terms there are).
Number of non-zero terms Name Example
0 zero polynomial 0
1 monomial 𝑥2
2 binomial 𝑥2 + 1
3 trinomial 𝑥2 + 𝑥 + 1
If a polynomial has only one variable, then the terms are usually written either from highest degree to lowest degree ("descending powers") or from lowest degree to highest degree ("ascending powers"). A polynomial in 𝑥 of degree n then takes the general form
𝑐𝑛𝑥𝑛 + 𝑐𝑛−1𝑥𝑛−1 + ⋯ + 𝑐2𝑥2 + 𝑐1𝑥 + 𝑐0
where
cn ≠ 0, cn-1, ..., c2, c1 and c0 are constants, the coefficients of this polynomial.
Here the term cnxn is called the leading term and its coefficient cn the leading coefficient; if the leading