6-5
Finding Real Roots of
Polynomial Equations
To find the roots of polynomials, completely
factor the polynomial and then set each factor
equal to zero.
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
4x4(x2 + x – 6) = 0
4x4(x + 3)(x – 2) = 0
Factor out the GCF, 4x4.
Factor the quadratic.
4x4 = 0 or (x + 3) = 0 or (x – 2) = 0 Set each factor
equal to 0.
Solve for x.
x = 0, x = –3, x = 2
Multiplicity of 4
The roots are 0, –3, and 2.
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 1A Continued
Check Use a graph. The
roots appear to be
located at x = 0, x = –3,
and x = 2. 
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 1B: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x4 + 25 = 26x2
x4 – 26 x2 + 25 = 0
(x2 – 25)(x2 – 1) = 0
Set the equation equal to 0.
Factor the trinomial in
quadratic form.
(x – 5)(x + 5)(x – 1)(x + 1) Factor the difference of two
squares.
x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0
x = 5, x = –5, x = 1 or x = –1
The roots are 5, –5, 1, and –1.
Holt Algebra 2
Solve for x.
6-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1a
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) = 0
Set each factor
equal to 0.
Solve for x.
x = 0, x = 5
Multiplicity of 2
The roots are 0 and 5.
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1b
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.
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1b
Solve the polynomial equation by factoring.
3x5 + 81x2 = 0
3x2(x3 + 27) = 0
Factor the GCF.
3x2(x + 3)(x2 – 3x + 9) = 0
Factor by sum of cubes.
3x2 = 0, x + 3 = 0, or x2 – 3x + 9 = 0
x = 0, x = -3, or x =
Multiplicity of 2
The roots are 0, -3,
Holt Algebra 2
Quadratic formula
3±3 3𝑖
2
3+3 3𝑖
,
2
and
3−3 3𝑖
.
2
6-5
Finding Real Roots of
Polynomial Equations
Each polynomial will have the same number of
roots as its degree. The roots may be real,
imaginary, double, triple (etc.), or some
combination.
The multiplicity of a root is the number of times
a number is a factor of P(x).
When a real root has even multiplicity, the graph
of y = P(x) touches the x-axis but does not cross
it.
When a real root has odd multiplicity greater than
1, the graph “bends” as it crosses the x-axis.
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
You cannot always determine the multiplicity of a
root from a graph. It is easiest to determine
multiplicity when the polynomial is in factored
form.
Holt Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Determine the multiplicity of each root for
the given graphs.
A. Degree 4 Poly
x = -1
x = 3 multiplicity of 3
Holt Algebra 2
B. Degree 4 Poly
x = -7 multiplicity of 2
x=0
x=7
6-5
Finding Real Roots of
Polynomial Equations
Determine the multiplicity of each root for
the given graphs.
A. Degree 3 Poly
x = -3
Two non-real zeros
Holt Algebra 2
B. Degree 6 Poly
x = -2 multiplicity of 2
x=0
x = 5 multiplicity of 3
6-5
Finding Real Roots of
Polynomial Equations
HW pg. 442
#’s 8-10, 21,22,24,26,29
Holt Algebra 2