Presentation on theme: "6.7 Using the Fundamental Theorem of Algebra"— Presentation transcript:
1 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?
2 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.
3 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, rational2x −1 = 0x2 −2 = 0x3 − 1 = 0(x −1)(x2 + x + 1), x = 1 and use Quadratic formula for
4 Solve the Polynomial Equation. x3 + x2 −x − 1 = 0Notice that −1 is a solution two times. This is called a repeated solution, repeated zero, or a double root.1 −1 −11121121x2 + 2x + 1(x + 1)(x + 1)x = −1, x = −1, x = 1
5 Finding the Number of Solutions or Zeros x+3x3 + 3x2 + 16x + 48 = 0(x + 3)(x2 + 16)= 0x + 3 = 0, x = 0x = −3, x2 = −16x = − 3, x = ± 4ix2x33x216x48+16
6 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, ±1/1Synthetic divisionx3 + 6x2 +12x + 8Zeros: −2,−2,−2, 0
7 Finding the Zeros of a Polynomial Function Find all the zeros of f(x) = x5 − 2x4 + 8x2 − 13x + 6Possible rational zeros: ±6, ±3, ±2, ±11 − −11−1−17−61−1−17−6−2−26−1061−35−311−231−23x2 −2x + 3Use quadratic formula
8 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.
9 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
10 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.
11 Assignment is p. 369,15-29 odd, oddShow your work
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