Warm - Up Perform the operation and write the result in standard form Use the quadratic formula to solve the quadratic equation

Assignment p. 266 # 11, 12, 16, 24, 26 Announcements Test Corrections Tomorrow After School Notebook Quiz Next Week

2.6 Fundamental Theorem of Algebra How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function and find all zeros of polynomial functions, including complex zeros How to find conjugate pairs of complex zeros How to find zeros of polynomials by factoring  

The Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n, where n > 0, then f has at least one zero in the complex number system. Does it have to cross the x-axis to be a zero?

Linear Factorization Theorem. The highest degree of a polynomial is how many complex zeros we should have. How many zeros for the following? x3 + 2x 3 3x2 + 2x4 + x + 5 4 7x4 - 2x5 + x6 – 4x 6

Find all the zeros of the function and write the polynomial as a product of linear factors Solve x3 + 6x – 7 = 0 Possible zeros: P = ± 7, ±1 Q = ±1 (p/q) = ± 7, ±1 How do we find the possible rational zeros? Factor Theorem – please test the four possibilities X = 1

Synthetic Division to simplify x3 + 6x – 7, x = 1 6 -7 7 Now, what do we do once we have a quadratic equation?

Using the Quadratic Formula

So…what are the roots? Using the factor theorem, synthetic division, and the quadratic formula, we have: x3 + 6x – 7 =

Steps for finding the zeros Use the Rational root test to find the possible rational zeros Use the factor theorem to test the possible rational zeros Use synthetic division until you find a quadratic equation Use the quadratic formula to find the complex zeros

Example: find all the zeros f(x) = x4 - 3x3 + x – 3 Use the Rational root test to find the possible rational zeros P = ±3, ±1 Q =± 1 (p/q) = ±3, ±1

Find the zeros f(x) = x4 - 3x3 + x – 3 Which ones work? X = -1, 3 Use the factor theorem to test the possible rational zeros (p/q) = ±3, ±1 f(x) = x4 - 3x3 + x – 3 Which ones work? X = -1, 3

Use synthetic division until you find a quadratic equation X = -1 1 -3 -1 4 -4 3

Use synthetic division until you find a quadratic equation X = 3 1 -4 4 -3 3 -1

Use the quadratic formula to find the complex zeros

Again, what are our zeros? f(x) = x4 - 3x3 + x – 3 Highest degree tells us there must be 4 zeros

Warm - Up Find all the zeros of the function and write the polynomials as a product of linear factors

Announcements p. 266 # 38 – 44 even, 51 – 55 odd Assignment p. 266 # 38 – 44 even, 51 – 55 odd Test Corrections today after school Notebook Quiz Next Week

Conjugate Pair whenever a + bi is a zero of f, a – bi is also a zero of f and vice versa. Example: Note that in Example 1, the two complex zeros were conjugates.

Name those conjugate pairs!!!

Find a polynomial function with integer coefficients that has the given zeros Conjugates Factor notation Rearrange FOIL Combine like terms Simplify

Find a fourth degree polynomial function with real coefficients that has 0, 1, and i as zeros. How do we write the three initial zeros? f(x) = x(x – 3)(x – i) What is the fourth zero? (remember that complex numbers come in conjugates) (x + i) f(x) = x(x – 1)(x – i)(x + i) = (x2 – x)(x2 – 1). = x4 – x3 - x2 + x.

Find a polynomial function with integer coefficients that has the given zeros

Warm Up How do we find the conjugate pair of a complex number? How do we multiply the factored form of complex numbers?

Announcements

Factoring a Polynomial Factor f(x) = x4 –12x2– 13 a) factor without square roots (x2 – 13)(x2 + 1) b) factor with square roots. c) factor completely

Finding the zeros of a polynomial Find all zeros of f(x) = x4 – 4x3 + 12x2 + 4x – 13 given that (2 + 3i) is a zero. Knowing about conjugates, what must be another zero? Since 2 + 3i is a zero, 2 – 3i is also a zero. That means that x2 – 4x + 13 is a factor of f(x).

Finding the zeros Given x2 – 4x + 13 is a factor of f(x) = x4 – 4x3 + 12x2 + 4x – 13 What type of division could we use to find the other factors? Long Division

Find the remaining zeros

…and so the zeros are? Zeros of f(x) = x4 – 4x3 + 12x2 + 4x – 13 After using complex conjugates, long division and factoring All the zeros of f are –1, 1, 2 + 3i, 2 – 3i.