Bellwork

iii

ii

32)5)(32()2

24)1 2

Perform the operation and write the result in standard from ( a + bi)

Last Nights Homework• 15. 11-i• 17. 7 - 3√2 i• 20. 17 + 18i• 30. 6 – 22i• 31. 12 + 30i• 34. 21-3√10 i + 7√5 i + √50• 47. -6i• 49. (16 + 20i)41• 59. -1 + 6i• 66. -1/8• 11-14. Will graph

2.5 Real and Complex Zeros of Polynomial Functions-How to find all zeros of polynomial functions, including complex zeros?

Every nth degree polynomial has precisely n zeros.

96)()

2)()

2

xxxfb

xxfa

Example 1: Find all the real and complex roots.

Example 1: Find all roots, complex and real. xxxfc 4)() 3 1)() 4 xxfd

Example 2: Find all zeros, real and complex.

16124)() 23 xxxxfa Start by finding zeroes in the calculator, use synthetic division, and then quadratic formula for the rest.

Example 2: Find all zeros, real and complex.

1812113)() 23 xxxxfb Start by finding zeroes in the calculator, use synthetic division, and then quadratic formula for the rest.

Complex zeros occur in conjugate pairs.

• Let f(x) be a polynomial function that has real coefficients. If a + bi, where b is not 0, is a zero of the function, the conjugate a – bi is also a zero of the function.

)3)(3)(1)(1()( ixixxxxf

918102)( 234 xxxxxf

Example 3: Find a fourth degree polynomial function, with real coefficients, that has -1, -1 and 3i as zeros.

Tonight’s Homework• Pg 187 #17, 19, 29, 38(just find all zeroes), 41, 43, 45, 65