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Using the Fundamental
Theorem of Algebra
What is the fundamental theorem of Algebra?
What methods do you use to find the zeros of a
polynomial function?
How do you use zeros to write a polynomial
function?
• German
mathematician Carl
Friedrich Gauss (17771855) first proved this
theorem. It is the
Fundamental Theorem
of Algebra.
If f(x) is a polynomial of degree n where n
> 0, then the equation f(x) = 0 has at least
one root in the set of complex numbers.
Solve each polynomial equation. State how
many solutions the equation has and
classify each as rational, irrational or
imaginary.
2x −1 = 0
x2
−2 = 0
x3 − 1 = 0
x = ½, 1 sol, rational
x  2, x   2
2
(x −1)(x2 + x + 1), x = 1
and use Quadratic
1 i 3
formula for
2
Solve the Polynomial Equation.
x3 + x2 −x − 1 = 0
1 1 1
1
1 2
1

1
−1 −1
2
1
1
0
x2 + 2x + 1
(x + 1)(x + 1)
x = −1, x = −1, x = 1
Notice that −1 is a
solution two times.
This is called a
repeated solution,
repeated zero, or a
double root.
Finding the Number of Solutions or Zeros
x3 + 3x2 + 16x + 48 = 0
(x + 3)(x2 + 16)= 0
x + 3 = 0, x2 + 16 = 0
x = −3, x2 = −16
x = − 3, x = ± 4i
Finding the Number of Solutions or
Zeros
f(x) = x4 + 6x3 + 12x2 + 8x
f(x)= x(x3 + 6x2 +12x + 8)
8 / = ±8 / , ±4 / , ±2 / ±1 /
1
1
1
1,
1
Synthetic division
x3 + 6x2 +12x + 8
1
6
12
8
Zeros: −2,−2,−2, 0
Finding the Zeros of a Polynomial Function
Find all the zeros of f(x) = x5 − 2x4 + 8x2 − 13x + 6
Possible rational zeros: ±6, ±3, ±2, ±1
1
−2
1
1
−2
1
0
8
−13
6
1 −1 −1
−1 −1 7
−2 6 −10
7
−6
6
−6
0
1 −3 5 −3
1 −2
3
1 −2 3
0
x2 −2x + 3
0
1,1,  2,1  i 2 ,1  i 2
Use quadratic formula
Graph of polynomial function
Recall:
Real zero: where the graph crosses the x-axis.
Repeated zero: where graph touches x-axis.
Using Zeros to Write Polynomial Functions
Write a polynomial function f of least degree that
has real coefficients, a leading coefficient of 1,
and 2 and 1 + i as zeros.
x = 2, x = 1 + i, AND x = 1 − i.
Complex conjugates always travel in pairs.
f(x) = (x − 2)[x − (1 + i )][x − (1 − i )]
f(x) = (x − 2)[(x − 1) − i ][(x − 1) + i ]
f(x) = (x − 2)[(x − 1)2 − i2 ]
f(x) = (x − 2)[(x2 − 2x + 1 −(−1)]
f(x) = (x − 2)[x2 − 2x + 2]
f(x) = x3 − 2x2 +2x − 2x2 +4x − 4
f(x) = x3 − 4x2 +6x − 4
• What is the fundamental theorem of Algebra?
If f(x) is a polynomial of degree n where n > 0,
then the equation f(x) = 0 has at least one
root in the set of complex numbers.
What methods do you use to find the zeros of a
polynomial function?
Rational zero theorem and synthetic division.
• How do you use zeros to write a polynomial
function?
If x = #, it becomes a factor (x - #).
Multiply factors together to find the
equation.