We saw previously that a polynomial equation of degree $n$n can have at most $n$n roots. One reason for the existence of fewer than $n$n roots is the possibility that some of the roots are the same. This occurs when the polynomial written in factored form has the same factor repeated a number of times.
Each root of a polynomial equation corresponds to a factor and the number of occurrences of the factor corresponding to a particular root is called the multiplicity of the root.
Example 1
The cubic equation $(x-1)^2(x+3)=0$(x−1)2(x+3)=0 has two distinct roots: $x=1$x=1 and $x=-3$x=−3. The root $x=1$x=1 has multiplicity $2$2.
The quadratic equation $x^2-6x+9=0$x2−6x+9=0 has a single root of multiplicity $2$2, also called a doubled root. This can be seen when the equation is written in the form $(x-3)^2=0$(x−3)2=0.
The quartic equation $x^4-2x^3+2x-1=0$x4−2x3+2x−1=0 has two roots: $x=1$x=1 and $x=-1$x=−1. The root $x=1$x=1 has multiplicity $3$3. The equation is equivalent to $(x-1)^3(x+1)=0$(x−1)3(x+1)=0.
If the functions that appear on the left of each of these equations are graphed, we observe that when the multiplicity of a zero of a function is greater than one, the tangent at the zero has zero slope. That is, the tangent coincides with the horizontal axis at the point where the function value is zero.
The following graph illustrates this using the three examples given above.
We can show this tangent property using differential calculus.
Example 2
Consider the function $f(x)=(x-\alpha)^mP(x)$f(x)=(x−α)mP(x). It has a zero $x=\alpha$x=α with multiplicity $m$m. The function $P(x)$P(x) is any polynomial which may or may not have other zeros.
We differentiate to obtain $f'(x)=m(x-\alpha)^{m-1}P(x)+(x-\alpha)^mP'(x)$f′(x)=m(x−α)m−1P(x)+(x−α)mP′(x).
At $x=\alpha$x=α, the derivative and hence the slope is zero. This will be true for all $m>1$m>1.
Practice Questions
Question 1
Which of the following equations has a root of multiplicity $4$4?
$\left(x+8\right)^4\left(x+3\right)=0$(x+8)4(x+3)=0
A$\left(x+8\right)\left(x+3\right)^3=0$(x+8)(x+3)3=0
B$\left(x+8\right)\left(x+3\right)=0$(x+8)(x+3)=0
C$\left(x+8\right)^3\left(x+3\right)=0$(x+8)3(x+3)=0
D
Question 2
Consider the graph of the cubic polynomial $P\left(x\right)$P(x) below. It has turning points at $x=1$x=1 and $x=3$x=3, and a single point of inflection at $x=2$x=2.
Complete the following sentences.
The root $x=1$x=1 of $P\left(x\right)=0$P(x)=0 has multiplicity $\editable{}$.
The root $x=4$x=4 of $P\left(x\right)=0$P(x)=0 has multiplicity $\editable{}$.
Question 3
Consider the following polynomial, which has a root of multiplicity $2$2.
$P\left(x\right)=x^3-x^2-8x+12$P(x)=x3−x2−8x+12.
Find the first derivative of $P\left(x\right)$P(x).
Solve $P'\left(x\right)=0$P′(x)=0.
Find the root of multiplicity $2$2 of $P\left(x\right)=0$P(x)=0, ensuring working is shown to justify your answer.
Let $x=a$x=a be the other root of $P\left(x\right)=0$P(x)=0. Find the value of $a$a.