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9.9 The Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

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Page 1: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

9.9 The Fundamental

Theorem of Algebra

Page 2: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

The Fundamental Theorem of Algebra

• Every polynomial equation with complex coefficients and positive degree n has exactly n complex roots.

• You may have to count the same number more than once if it is a root.

double

Page 3: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

• Theorem: If a polynomial equation with real coefficients has as a root (a and b real), then is also a root. In other words, imaginary roots come in pairs -

.

• Just like we did with quadratic equations, we can also write the equation of any polynomial from its roots.

a +bi a−bi

complex conjugates

Page 4: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Find the polynomial equation of least degree having the given roots.

1. 2, 1, -4

Page 5: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Find the polynomial equation of least degree having the given roots.

2. 2, 1, 1+ i

Page 6: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Given the following root(s) for the polynomial equation, find

the remaining roots.

3. x3 −3x2 + x−3 =0; 3

Page 7: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Given the following root(s) for the polynomial equation, find

the remaining roots.

4. x3 −2x2 −4x−16 =0; −1+ i 3

Page 8: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Given the following root(s) for the polynomial equation, find

the remaining roots.

5. x4 +11x2 +18 =0; −3i

Page 9: 9.9 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra Every polynomial equation with complex coefficients and positive degree n has

Given the following root(s) for the polynomial equation, find

the remaining roots.

6. x4 + 4x3 + 2x2 −12x−15 =0; −2 −i


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