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**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?

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German mathematician Carl Friedrich Gauss ( ) 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.

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**Solve each polynomial equation**

Solve each polynomial equation. State how many solutions the equation has and classify each as rational, irrational or imaginary. x = ½, 1 sol, rational 2x −1 = 0 x2 −2 = 0 x3 − 1 = 0 (x −1)(x2 + x + 1), x = 1 and use Quadratic formula for

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**Solve the Polynomial Equation.**

x3 + x2 −x − 1 = 0 Notice that −1 is a solution two times. This is called a repeated solution, repeated zero, or a double root. 1 −1 −1 1 1 2 1 1 2 1 x2 + 2x + 1 (x + 1)(x + 1) x = −1, x = −1, x = 1

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**Finding the Number of Solutions or Zeros**

x +3 x3 + 3x2 + 16x + 48 = 0 (x + 3)(x2 + 16)= 0 x + 3 = 0, x = 0 x = −3, x2 = −16 x = − 3, x = ± 4i x2 x3 3x2 16x 48 +16

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**Finding the Number of Solutions or Zeros**

f(x) = x4 + 6x3 + 12x2 + 8x f(x)= x(x3 + 6x2 +12x + 8) 8/1= ±8/1, ±4/1, ±2/1, ±1/1 Synthetic division x3 + 6x2 +12x + 8 Zeros: −2,−2,−2, 0

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**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 − − 1 1 −1 −1 7 −6 1 −1 −1 7 −6 −2 −2 6 −10 6 1 −3 5 −3 1 1 −2 3 1 −2 3 x2 −2x + 3 Use quadratic formula

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**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.

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**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 − 4 f(x) = x3 − 4x2 +6x − 4

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**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.

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Assignment is p. 369, 15-29 odd, odd Show your work

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