6.7 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?

How do you use zeros to write a polynomial function?

• 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.

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

2,22 xx

x = ½, 1 sol, rational

(x −1)(x2 + x + 1), x = 1 and use Quadratic formula for

2

31 i

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.

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

Finding the Number of Solutions or Zeros

f(x) = x4 + 6x3 + 12x2 + 8xf(x)= x(x3 + 6x2 +12x + 8)8/1= ±8/1, ±4/1, ±2/1, ±1/1

Synthetic division x3 + 6x2 +12x + 81 6 12 8

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

21,21,2,1,1 ii

Graph of polynomial function

Turn to page 367 in your book.

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 − 4f(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 (6.6) 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.

Assignment is p. 369, 15-29 odd, 35-43 odd

Show your work