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

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