Download 6.7 Using the Fundamental Theorem of Algebra

2.7 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?
Find the number of solutions or
zeros
a. How many solutions does the equation
x3 + 5x2 + 4x + 20 = 0 have?
SOLUTION
Because x3 + 5x2 + 4x + 20 = 0 is a
polynomial
equation of degree 3,it has three solutions.
(The solutions are – 5, – 2i, and 2i.)
Find the number of
solutions or zeros
1. How many solutions does the
equation
x4 + 5x2 – 36 = 0 have?
ANSWER 4
2.
How many zeros does the function
f (x) = x3 + 7x2 + 8x – 16 have?
ANSWER 3
• German mathematician
Carl Friedrich Gauss
(1777-1855) 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.
If f(x is a polynomial of degree n where n>0, then the
equation f(x) has exactly n solutions provided each
solution repeated twice is counted as 2 solutions,
each solution repeated three times is counted as 3
solutions, and so on.
Solve the Polynomial Equation.
x3 + x2 −x − 1 = 0
1 1 1
1
1 2
1

1
−1 −1
2
1
1
0
Notice that −1 is a
solution two times.
This is called a
repeated solution,
repeated zero, or a
double root.
x2 + 2x + 1
(x + 1)(x + 1)
x = −1, x = −1, x = 1
http://my.hrw.com/math06_07/nsmedia/t
ools/Graph_Calculator/graphCalc.html
Finding the Number of Solutions or Zeros
x
x3 + 3x2 + 16x + 48 = 0
x2
(x + 3)(x2 + 16)= 0
x + 3 = 0, x2 + 16 = 0
x = −3, x2 = −16
+16
x = − 3, x = ± 4i
+3
x3
3x2
16x
48
Find the zeros of a polynomial equation
Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14.
SOLUTION
STEP 1 Find the rational zeros of f. Because f is a
polynomial function of degree 5, it has 5
zeros. The possible rational zeros are + 1,
+ 2, + 7, and + 14. Using synthetic division,
you can determine that – 1 is a zero
repeated twice and 2 is also a zero.
STEP 2 Write f (x) in factored form. Dividing f (x)
by its known factors x + 1, x + 1, and x – 2
gives a quotient of x2 – 4x + 7. Therefore:
f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)
STEP 3
Find the complex zeros of f . Use the quadratic
formula to factor the trinomial into linear factors.
f(x) = (x + 1)2(x – 2) x – (2 + i
3 )
x – (2 – i
3 )
ANSWER
The zeros of f are – 1, – 1, 2, 2 + i
3 , and 2 – i
3.
Finding the Number of Solutions or
Zeros
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 or complete the square
• 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 (2.6) and synthetic
division.
Assignment is:
Page 141, 3-9 all, 11-17 odd
Show your work
NO WORK NO CREDIT