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?
ANSWER4
2. How many zeros does the function f (x) = x3 + 7x2 + 8x – 16 have?
ANSWER3
• 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 = 01
1
1 1 −1 −11
x2 + 2x + 1
(x + 1)(x + 1)
x = −1, x = −1, x = 1
1 2 1
1 2 1 0
Notice that −1 is a solution two times. This is called a repeated solution, repeated zero, or a double root.
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html
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
x3 3x2
16x 48
x2
+16
x +3
Find the zeros of a polynomial equationFind 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 )
The zeros of f are – 1, – 1, 2, 2 + i 3 , and 2 – i 3.
ANSWER
Finding the Number of Solutions or Zeros
Zeros: −2,−2,−2, 0
Finding the Zeros of a Polynomial FunctionFind all the zeros of f(x) = x5 − 2x4 + 8x2 − 13x + 6
Possible rational zeros: ±6, ±3, ±2, ±1
1 −2 0 8 −13 61
1
1
−1
−1
−1
7
7
−6
0−2
1
−2
−3
6
5
−10
−1
−6
−3
6
01
11
−2−2
330
x2 −2x + 3
Use quadratic formula or complete the square
21,21,2,1,1 ii
• 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 workNO WORK NO CREDIT
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