Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

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Presentation transcript:

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

COMPLEX NUMBER SYSTEM Natural Whole Integers RationalIrrational REAL NUMBER SYSTEM Imaginary Numbers

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. True for integral, rational, irrational and imaginary coefficients

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. True for integral, rational, irrational and imaginary coefficients

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. We are going to get integral, rational, irrational and imaginary ZEROS

Fundamental Theorem of Algebra Another Theorem

is a polynomial function of positive degree with complex coefficients, then P(x) has n linear factors.

Last Theorem Corrollary

Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

Find the Integral ZEROS only Determine the zeros of the polynomial

13-3 NO Integral ZEROS

13-3

Determine the zeros of the polynomial

Determine the zeros of the polynomial Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

13-3 That’s why there was only ONE x-intercept

Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

Complex Conjugate Theorem If a complex number, a + bi, is a zero of a polynomial function with real coefficients, then its conjugate, a – bi, is also a zero of the polynomial

Determine all the zeros of the polynomial given one of the zeros i Leave more space between the numbers than you usually do

Determine all the zeros of the polynomial given one of the zeros i i i

Determine all the zeros of the polynomial given one of the zeros i i i

Determine all the zeros of the polynomial given one of the zeros i i i i i i

Determine all the zeros of the polynomial given one of the zeros i i i i i

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i -35 – 6i -39

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i -39

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i -39

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i -39

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i Complex Conjugate Theorem

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 3 9 – 6i 2

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i 32– 3

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3 Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

Determine all the zeros of the polynomial given one of the zeros i i i i i -35 – 6i – 6i – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

Conjugate Radical Theorem If, is a zero of a polynomial function with rational coefficients, where a and b are rational, but is irrational, then the conjugate number is also a zero of the polynomial

Determine all the zeros of the polynomial given one of the zeros –1–1 Leave more space between the numbers than you usually do

Determine all the zeros of the polynomial given one of the zeros –1–1

–1–1

–1–1

–1–1

–1–1

–1–1 1 0

–1–1 1 0

–1–1 1 0

–1–1 1 0 Conjugate Radical Theorem

Determine all the zeros of the polynomial given one of the zeros –1–

–1–

–1–

–1–

–1–

–1– Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

Determine all the zeros of the polynomial given one of the zeros –1–

Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

Determine all the zeros of the polynomial given one of the zeros. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

Determine all remaining zeros of each polynomial given one of the zeros