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Unit 3.3Polynomial Equations
Continued
Objectives
Divide polynomials with synthetic division
Combine graphical and algebraic methods to solve
polynomial equations
Use the Fundamental Theorem of Algebra to find
the number of complex solutions of a polynomial
equation
Find the complex solutions of polynomial
equations
Division of Polynomials; Synthetic Division
Suppose f(x) is a cubic function and we know that a is a
solution of f (x) = 0. We can write (x - a) as a factor of f
(x), and if we divide this factor into the cubic function, the
quotient will be a quadratic factor of f (x). If there are
additional real solutions to f(x) = 0, this quadratic factor
can be used to find the remaining solutions.
4
3
Divide 𝑥 + 6𝑥 − 5𝑥 + 4 by x + 2
Long Division
Synthetic Division
Example
If x = 2 is a solution of x3 + 6x2 – x – 30 = 0, find the
remaining solutions.
Solution
Example (cont)
Thus the quotient is ________________, with remainder 0, so
x – 2 is a factor and
Y
18
12
6
X
-7
-6
-5
-4
-3
-2
-1
0
-6
-12
-18
The solutions are x = ___, __, and ___.
-24
-30
-36
1
2
3
Multiplicity
A graph touches but does not cross the x-axis at
a zero of even multiplicity, and it crosses the xaxis at a zero of odd multiplicity.
Example
The weekly profit for a product is P(x) = – 0.1x3 + 11x2 – 80x – 2000
thousand dollars, where x is the number of thousands of units
produced and sold. To find the number of units that gives break-even,
graph the function using a window representing up to 50 thousand
units and find one x-intercept of the graph.
We use the viewing window
[0, 50] by [–2500, 8000].
Example (cont)
b. Use synthetic division to find a quadratic factor of P(x).
Solution
Example (cont)
c. Find all of the zeros of P(x).
Solution
Example:
Determine all possible rational solutions:
3𝑥 3 + 3𝑥 2 − 9𝑥 + 12 = 0
Example
Solve the equation 2x4 + 10x3 + 13x2 – x – 6 = 0.