Understanding Engineering Mathematics

2.2.6 Roots and factors of a polynomial

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Shortly, we will be looking at more powerful methods for factorising polynomials. First, I
want to clarify something that often confuses those new to algebra. This is the distinction
between factors, roots and solutions of polynomials or polynomial equations. Remember
the difference between an algebraicexpressionand an algebraicequationreferred to in
Section 2.2.1? In particular, apolynomialis an expression such as

x^2 +x− 2

whereas the correspondingpolynomial equationwould be

x^2 +x− 2 = 0

Now if we proceed to solve this equation by factorising:

x^2 +x− 2 ≡(x− 1 )(x+ 2 )

the expressionsx−1,x+2 are factors of thepolynomial expressionalone, independently
of the equation. The last identity is simply an alternative way of writingx^2 +x−2in
terms of these factors. This is of course helpful insolvingthe correspondingpolynomial
equation:

x^2 +x− 2 ≡(x− 1 )(x+ 2 )= 0

Remembering that for real numbersab=0 means one or both ofa,bmust be zero (41

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this factorised form tells us that the solutions to the equation can be obtained by putting
the factorsx−1,x+2 equal to zero. This leads to thesolutionsof the equation

x= 1 ,x=− 2

These values ofx, which make the polynomialx^2 +x−2 zero, are called therootsor
zerosof the polynomial. So, ifx−1isafactorof the quadratic expressionx^2 +x− 2
thenx=1isasolutionof the quadratic equationx^2 +x− 2 =0.
In general, ifx−ais a factor of a polynomialpn(x)thenx=awill be a solution of
the polynomial equationpn(x)=0. The solutionx=ais often referred to as arootof
the polynomial. This idea is the basis of thefactor theorem:

Ifp(x)is a polynomial and ifx=ais a root of the polynomial, i.e.p(a)=0, then
x−ais a factor ofp(x). On the other hand, ifx−ais a factor ofp(x)then clearly
p(a)=0.

This result is fairly obvious from the fact that ifp(x)has a factorx−athen it can be
written asp(x)=(x−a)q(x)whereq(x)is another polynomial. Then clearlyp(a)=0.
We can use the factor theorem in factorising more complicated polynomials. While
we know from the examplex^2 +1 that not all polynomials can be factorised into linear
factorsx−a,itcanbe shown thatany polynomial with real coefficients can always be
factorised into linear and/or quadratic factors with real coefficients.Itmaythenbe
possible to find the linear factors by trial and error using the factor theorem, if the roots
are simple numbers, easy to spot.
Another useful result is that if the coefficient of the term of highest degree is unity the
roots of a polynomial may be factors of the constant term, which gives us some clues
about what the factors might be.