6.7 Using the Fundamental Theorem of Algebra

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Presentation transcript:

6.7 Using the Fundamental Theorem of Algebra

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.

Finding the Number of Solutions or Zeros State the number of solutions and tell what they are. x2 – 14x + 49 = 0 x4 + 3x3 – 8x2 – 22x – 24 = 0 x3 – 3x + 52 = 0

Finding all the zeros of the polynomial function Find all the zeros. f(x) = x3 + x2 – x + 15 f(x) = x3 + 5x2 – 6 f(x) = x3 – x2 + 4x – 4

Finding all the zeros of the polynomial function Find all the zeros. f(x) = x3 + x2 + 9x + 9 f(x) = x4 + 5x3 + 10x2 + 20x + 24 f(x) = x4 – 6x2 + 5

Using Zeros to Write Polynomial Functions Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1 and 1, -2 + i, and -2 + i as zeros. Write a polynomial function of least degree that has reals coefficients, a leading coefficient of 1, and 5, 2i, and -2i as zeros.

Approximate the Real Zeros f(x) = x3 – 4x2 – 5x + 14 f(x) = x3 – 3x2 – 2x + 6

Story Problems A rectangular piece of sheet metal is 10in wide. Squares of side length x are cut from the corners and the remaining piece is folded to make a box. The volume of the box is modeled by V(x) = 4x3 – 4x2 + 100x. What size square can be cut from the corners to give a box with a volume of 25in3?