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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. State how many solutions the equation has and classify each as rational, irrational or imaginary. 2x 1 = 0 x 2 2 = 0 x 3 1 = 0 x = ½, 1 sol, rational (x 1)(x 2 + x + 1), x = 1 and use Quadratic formula for

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Solve the Polynomial Equation. x 3 + x 2 x 1 = x 2 + 2x + 1 (x + 1)(x + 1) x = 1, x = 1, x = Notice that 1 is a solution two times. This is called a repeated solution, repeated zero, or a double root.

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Finding the Number of Solutions or Zeros x 3 + 3x x + 48 = 0 (x + 3)(x )= 0 x + 3 = 0, x = 0 x = 3, x 2 = 16 x = 3, x = ± 4i x3x3 3x 2 16x48 x2x2 +16 x+3

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Finding the Number of Solutions or Zeros f(x) = x 4 + 6x x 2 + 8x f(x)= x(x 3 + 6x 2 +12x + 8) 8 / 1 = ± 8 / 1, ± 4 / 1, ± 2 / 1, ± 1 / 1 Synthetic division x 3 + 6x 2 +12x Zeros: 2,2,2, 0

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Finding the Zeros of a Polynomial Function Find all the zeros of f(x) = x 5 2x 4 + 8x 2 13x + 6 Possible rational zeros: ±6, ±3, ±2, ± x 2 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 i 2 ] f(x) = (x 2)[(x 2 2x + 1 (1)] f(x) = (x 2)[x 2 2x + 2] f(x) = x 3 2x 2 +2x 2x 2 +4x 4 f(x) = x 3 4x 2 +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, odd, odd Show your work

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