Roots of Polynomials
The number of roots (zeros, solutions) of a
polynomial is equal to the degree of the polynomial.
The number of turns (relative mins/maxs) is one less
than the degree of the polynomial.
P(x) = 4x3 – 5x + 2 has 3 roots, 2 turns.
R(x) = 15x5 – 1 has 5 zeros, 4 turns.
Some roots have a multiplicity meaning they are a
double, triple, or more root.
Single root: the line simply
passes through the x-axis
This graph has 3 single roots, 2
turns. It is degree 3.
Double root: the line
touches but does not go
through the x-axis
This graph has 2 double roots,
3 turns. It is degree 4.
Triple root: the line bends
as it goes through the xaxis
Counts as a double turn.
This graph has 2 triple roots, 5
turns. It has degree 5.
What kind(s) of roots does this graph have?
This graph has 2 double roots and 1 single
root, 4 turns. It is degree 5.
What kind(s) of roots does this graph have?
This graph has 1 double root and 1 triple root, it
has 4 turns. It is degree 5.
What kind(s) of roots does this graph have?
This graph has 2 single roots and 2 double
roots, it has 5 turns. It is degree 6.
Solve the polynomial equation by factoring.
4x5 + 4x4 – 24x3 = 0
Factor out the GCF, 4x4.
4x3(x2 + x – 6) = 0
4x3(x + 3)(x – 2) = 0
Factor the quadratic.
4x3 = 0 or (x + 3) = 0 or (x – 2) = 0
x = 0, x = –3, x = 2
Set each factor
equal to 0.
Solve for x.
Multiplicity of 3
The roots are 0 multiplicity of 3, –3, and 2.
Solve the polynomial equation by factoring.
2x3 – 20x2 = -50x
2x3 – 20x2 + 50x= 0
Write in standard form.
2x(x2 – 10x + 25) = 0
Factor out the GCF, 2x4.
2x(x – 5)(x – 5) = 0
Factor the quadratic.
2x = 0 or (x – 5)2 = 0
Set each factor
equal to 0.
x = 0, x = 5
Solve for x.
Multiplicity of 2
The roots are 0 and 5.
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
x3 – 2x2 – 25x + 50 = 0
Set the equation equal to 0.
x2(x – 2) – 25(x – 2) = 0
Factor by grouping.
(x2 – 25)(x – 2) = 0
(x + 5)(x – 5)(x – 2) = 0
Factor by diff of squares.
x + 5 = 0, x – 5 = 0, or x – 2 = 0
The roots are –5, 5, and 2.
Write a polynomial that has a zero at 4 with
double multiplicity and a zero at -1.
Mult. of 2
Mult. of 1
(x – 4)2
(x + 1)
Zero at 4
Zero at -1
(x – 4)2(x + 1)
Multiply the factors
(x2 – 8x + 16)(x + 1)
FOIL (x-4)(x-4)
Distribute to (x+1)
x3 – 7x2 + 8x + 16
Note: you do not have to multiply (simplify)
Write a polynomial that has a zero at 5 multiplicity of
2, a zero at 3, a zero at -3 with multiplicity 3, and a
zero at -4
Mult. of 2
Mult. of 1
Mult. of 3
Mult. of 1
(x – 5)2
(x – 3)
(x + 3)3
(x + 4)
Zero at 5
Zero at 3
Zero at -3
Zero at -4
P(x) = (x – 5)2(x – 3)(x + 3)3(x + 4) Multiply the factors
Determine the degree of the polynomial from its
graph.
This graph has 1
double root and 2
single roots, it has 5
turns. So it must be
at least degree 6
based on the turns.
If the polynomial is degree 6 but only has 4 roots
accounted for (1 double, 2 single), then where are
the other two roots?
They’re imaginary! We will find them tomorrow.