Understanding Engineering Mathematics

2.2.2 Polynomials

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Amonomialis an algebraic expression consisting of a single term, such as 3x, while a
binomialconsists of a sum of two terms, such asx+ 3 y.Apolynomialis an algebraic
expression consisting of a sum of terms each of which is a product of a constant and
one or more variables raised to a non-negative integer power. An example with a single
variable,x,is

x^3 − 2 x^2 +x+ 4

and the general form of a polynomial inxis written:

pn(x)=anxn+an− 1 xn−^1 +an− 2 xn−^2 +···+a 1 x+a 0

where theai,i= 0 , 1 ,...,nare given numbers called thecoefficientsof the polynomial.
We u s epn(x)to denote a polynomial inxof degreen. The notationaiis often used
when we have a list of quantities to describe –iis called thesubscript.Ifnis the highest
power that occurs, as in the above, and ifan =0, then we say that the polynomial is of
nth degree. An important property of a polynomial inxis that itexists(i.e. has a definite
value) for every possible value ofx.
A polynomial of degree zero is simply aconstant:

p 0 (x)=a 0

A polynomial of degree one:

p 1 (x)=a 1 x+a 0

is alinear polynomial,orlinear function.
A polynomial of degree two:

p 2 (x)=a 2 x^2 +a 1 x+a 0

is aquadratic polynomial,orquadratic function.
Similarly for cubic (degree 3), quartic (degree 4), quintic (degree 5),...,etc.
An equation of the form

pn(x)=anxn+an− 1 xn−^1 +···+a 1 x+a 0 = 0

is apolynomial equation inxof degreen.

Examples

(i) p(x)= 2 x+1isalinear polynomialorfunction, while 2x+ 1 = 0

is alinear equation, which givesx=−^12

(ii) p(x)=x^2 − 3 x+2isaquadratic function.x^2 − 3 x+ 2 =0is

a quadratic equation, which can be solved by factorising (see

Section 2.2.3):

x^2 − 3 x+ 2 =(x− 1 )(x− 2 )= 0