Precalculus: 3.3 Theory of Equations
Concepts: Multiplicity, n-Root Theorem, Conjugate Pairs Theorem, root/zero/x-intercept, Descartes’s
Rule of Signs
Definition (multiplicity): If the polynomial fhas (x−c)mas a factor but not (x−c)m+1, then cis a zero of f
of multiplicity m.
Example Find all the roots of f(x) = 9
4x2−21x+ 49.
Seeing that the 9
4= (3
2)2and 49 = 72, we might guess this factors as a perfect square, f(x) = ( 3
2x−7)2.
Let’s check if our guess is correct:
(3
2x−7)2= (3
2x−7)(3
2x−7)
=9
4x2−21
2x−21
2x+ 49
=9
4x2−21x+ 49
=f(x)
So since f(x)=(3
2x−7)2,fhas a root of c= 14/3 of multiplicity 2.
n-Root Theorem: If P(x) = 0 is a polynomial equation with real or complex coefficients and positive degree n,
then (including multiplicity) P(x) = 0 has nroots.
Conjugate Pairs Theorem: If P(x) = 0 is a polynomial equation with real coefficients and the complex number
c=a+bi is a root, then the complex conjugate ¯c=a−bi is also a root.
These theorems tell us that any polynomial of degree nwith real coefficients can be written as
P(x)=(x−c1)(x−c2)(x−c3). . . (x−cn) =
n
X
i=1
(x−ci),
where some of the ci∈Cmay be repeated, and complex roots appear in complex conjugate pairs.
Although we won’t be looking at sketching polynomials by hand until later, you may be using a calculator to sketch
polynomials to help you visualize examples, and there are some things to note about the sketches of polynomials.
When we look at a sketch of a polynomial of degree n, we may not see n x-intercepts due to two things:
•some of the zeros may be real, but have multiplicity greater than one, and
•some of the zeros may be complex.
How Multiplicity of zero c∈RAffects the Behaviour of f(x)
•If c∈Ris a zero of the polynomial fwith odd multiplicity, then the graph of fcrosses the xaxis at
x=c. This is because the function fwill change sign at x=c.
•If c∈Ris a zero of the polynomial fwith even multiplicity, then the graph of fdoes not cross the x
axis at x=c, but does touch the xaxis at x=c. This is because the function fwill not change sign at
x=c.
•If the multiplicity is greater than or equal to 2, the graph will be horizontal where it touches the x-axis.
Page 1 of 3
