2.6 Solving Polynomial Equations
Warm-Up: Solve x  27  0 . There will be 3 solutions.
3
3
2
2
Recall factoring for cubes: a  b  a  b  a  ab  b
3


Definition: Polynomial Function--A polynomial function of degree n in one variable is a
function of the form f  x   a n x n  a n 1 x n 1      a1 x  a 0 where a n  0 ; n is a
non-negative integer.
“Multiplicity”--When the solution of an equation (the zeros) is repeated, we say that the
solution gives multiple zeros. When a zero occurs k times, it is said to have
“multiplicy k.”
An example of Solving Polynomial Equations by Factoring:
Find the zeros of the polynomial function f x   2 x 5  8x 4  154 x 3 .
1.
2.
3.
4.
5.
Solve the equation f(x)=0.
Find the GCF.
Factor.
Apply the zero-product property.
Solve for x.
An example of Solving Polynomial Equations by Grouping:
Solve the polynomial equation 2 x 3  6 x 2  5 x  15.
1.
2.
3.
4.
5.
6.
Solve the equation f(x)=0.
Group terms.
Factor each group.
Identify the GCF.
Apply the zero-product property.
Solve for x.
An example of Solving Polynomial Equations in quadratic form:
Solve the polynomial equation 12t 4  5t 2  2  0.
1. Factor like a normal trinomial.
2. Apply the zero-product
property.
3. Solve for x.
Using your Calculator to estimate real zeros:
1. Enter the function in your calculator.
2. Zoom and Trace
3. Round to the requested degree.
Example: f (x)  2x 3  6x 2  x  4. Round to the nearest hundredth.
Can you find the relative maximum and relative minimum?
Applying this concept to a real situation:
Dr. Werner owns a rectangular fish tank with a square bottom whose height is half the
length of its side. The fish tank holds 4000 cu. in. of water at maximum capacity.
Find the dimensions of the tank.
Try These: Solve for x over the Complex numbers and state multiplicities. Remember, start
by setting the equation equal to zero.
1. x 3  x 2  9x  9
2.
x 4  36  5x 2
3x 6  24x 3  0
4.
28x 5  13x 4  6x 3  0
3.
Using your calculator functions:
5. f (x)  2x
intercepts).
6.
3
 5x 2  2x  4 . Find and relative extrema (minimums or maximums), find zeros (x-
f (x)  x 3  3x 2  x  1 . Find and relative extrema (minimums or maximums), find zeros (x-intercepts).