Degree of a Polynomial

Samia CHALLAL

6 Degree of a Polynomial

A polynomial
is an expression that can be written as a sum of terms of the form $\alpha x_1^{n_1} x_2^{n_2} \ldots x_m^{n_m}$ where $\alpha$ is a
constant and $x_1, x_2, \ldots, x_m$ are variables.

A monomial, a binomial, and a trinomial, are polynomials of one, two and three terms respectively.

The degree of a polynomial is the largest of the degrees of its individual terms.

The degree of a term in a polynomial is the sum of the exponents of its variables.

ex. degree of $\alpha x_1^{n_1} x_2^{n_2} \ldots x_m^{n_m}$ is equal to $n_1+n_2+\ldots+n_m$

The standard form of a polynomial of one variable $x$ is : $a_n x^n + a_{n-1} x^{n-1} + \ldots+ a_1 x + a_0$

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Exercise 1

Exercise 2

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The degree of a polynomial is the largest of the degrees of its individual terms.

The degree of a term in a polynomial is the sum of the exponents of its variables.

The degree of $\Big( {5} x^{{3}} y^{{6}} \Big) \Big( {5} x^{{5}} y^{{5}}\Big)^2$ is :

$={3} + {6} + 2\times ({5}) + 2\times ({5}) = 29$ .

Exercise 3

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The degree of

$\Big( {6} t - 3 q + q^{{5}} \Big) \Big( t^{{6}} - t^{{5}} q^{{6}} + q^{{5}} \Big)$

is equal to the degree of the term $q^{{5}} t^{{5}} q^{{6}}$ ; that is $= {5}+ {5} + {6} = 17$ .

Exercise 4

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The degree of

$\Big( {3} x + {4} y^2 + z^{{3}} \Big) \Big( - 5 x^{{5}} z^{{4}} y^{{3}} + y^{{4}} \Big) + x^{{3}} - {5} y^{{4}} z^{{3}} - {3} z^{{3}}$

is equal to the degree of the term $z^{{3}}x^{{5}} z^{{4}} y^{{3}}$ ; that is $= 3+ {5}+ {4} + {3} = 15$ .

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