In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see order of a polynomial (disambiguation)).
For example, the polynomial
7
x
2
y
3
+
4
x
−
9
,
{\displaystyle 7x^{2}y^{3}+4x-9,}
which can also be written as
7
x
2
y
3
+
4
x
1
y
0
−
9
x
0
y
0
,
{\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},}
has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term.
To determine the degree of a polynomial that is not in standard form, such as
(
x
+
1
)
2
−
(
x
−
1
)
2
{\displaystyle (x+1)^{2}-(x-1)^{2}}
, one can put it in standard form by expanding the products (by distributivity) and combining the like terms; for example,
(
x
+
1
)
2
−
(
x
−
1
)
2
=
4
x
{\displaystyle (x+1)^{2}-(x-1)^{2}=4x}
is of degree 1, even though each summand has degree 2. However, this is not needed when the polynomial is written as a product of polynomials in standard form, because the degree of a product is the sum of the degrees of the factors.
Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
The Homework Statement reads the question.
We have
$$
\langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right)
$$
If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows
$$...
The Gauss-Kronrod quadrature uses the zeros of the Legendre Polynomials of degree n and the zeros of the Stieltjes polynomials of degree n+1. These zeros are the nodes for the quadrature. For example using the Gauss polynomial of degree 7, you will need the Stieltjes of degree 8 and both makes...
Homework Statement
A monic polynomial is a polynomial which has leading coefficient 1. Find the real, monic polynomial of the lowest possible degree which has zeros −1−2i,−2i and i. Use z as your variable.
The Attempt at a Solution
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Would I just expand the zeros giving me...
Homework Statement
Evaluate the definite integral below numerically (between limits -1 and 1) using a couple of numerical methods, including Gauss-Legendre quadrature - and compare results.
Homework Equations
$$ \int{(1-x^2)^\frac{1}{2}} dx $$
"Gauss quadrature yields the exact integral if φ...