6.7 Using the Fundamental Theorem of Algebra

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Presentation transcript:

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 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

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

Finding the Number of Solutions or Zeros x +3 x3 + 3x2 + 16x + 48 = 0 (x + 3)(x2 + 16)= 0 x + 3 = 0, x2 + 16 = 0 x = −3, x2 = −16 x = − 3, x = ± 4i x2 x3 3x2 16x 48 +16

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 1 6 12 8 Zeros: −2,−2,−2, 0

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 −2 0 8 −13 6 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

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 − 4 f(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