Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.

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Presentation transcript:

Using the Fundamental Theorem of Algebra 6.7

Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number of zeros of polynomial function. -Use technology to approximant the real zeros of polynomial function.

Warm-up

Go Over Quiz

Review Find zeros of functions, solutions to polynomial equations A. If possible, factor B.If not factorable, find one root by using the Rational Root Theorem (p/q), use synthetic division and then factor the remaining quadratic. C.Finally, set unsolved factors to zero and solve

Using the fundamental theorem to determine the number of roots. 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.

We know: the degree of the equation tells you the number of solutions imaginary solutions come in pairs

New info: if an imaginary number is a zero, then it’s conjugate is also a zero – so, if 4 + 3i is a solution, then 4 – 3i is also a solution

Example 1 State the number of solutions and tell what they are. x 2 – 14x + 49 = 0

Example 2: Solve. x 4 + 3x 3 – 8x 2 – 22x – 24 = 0 How many solutions are there?4 Use what you did in 6.6 to solve for them. List the possible rational roots. Use your calculator to pick ones to test. Synthetically divide until you get a quadratic Factor or use the quadratic formula.

Writing polynomial functions from zeroes Example 1: Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 and has the roots 3, 2, –4. 1. Rewrite the zeros as factors.2. Multiply/Expand

Example 2 Write a polynomial function of least degree that has real coefficients and a leading coefficient of 3 and has the roots 5, 2i.

Example 3 Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 and has the roots 1 and 2 + i.

Closure

Homework Section 6.7 Page 369 – 370 #17 – 47 every 3