Warm-Up: January 9, 2012

Homework Questions?

Zeros of Polynomial Functions Section 2.5

Rational Zero Theorem If f(x) is a polynomial with integer coefficients, then every possible rational zero is given by: Factors of the constant term Possible rational zeros = Factors of the leading coefficient

Example 1 (like HW #1-8) List all possible rational zeros for

You-Try #1 (like HW #1-8) List all possible rational zeros for

Finding Zeros of Polynomial Functions Use the rational zero theorem to find all the possible rational zeros. Use a guess-and-check method and synthetic division to find a zero Use the successful synthetic division to get a lower- degree polynomial that can then be solved to find the remainder of the zeros.

Example 3 (like HW #9-22) Find all rational zeros of

You-Try #3 (like HW #9-22) Find all rational zeros of

Warm-Up: January 10, 2012 Find all rational zeros of

Homework Questions?

Example 4 (like HW #9-22) Solve

You-Try #4 (like HW #9-22) Solve

Linear Factorization Theorem An nth-degree polynomial has n complex roots c1, c2, …, cn Each root can be written as a factor, (x-ci) An nth-degree polynomial can be expressed as the product of a nonzero constant and n linear factors:

Example 5 (like HW #23-28) Factor the polynomial as the product of factors that are irreducible over the rational numbers as the product of factors that are irreducible over the real numbers in completely factored form, including complex (imaginary) numbers

You-Try #5 (like HW #23-28) Factor the polynomial as the product of factors that are irreducible over the rational numbers as the product of factors that are irreducible over the real numbers in completely factored form, including complex (imaginary) numbers

Warm-Up: January 11, 2012 Find all zeros of Hint: Start by factoring as we did in Example 5.

Homework Questions?

Finding a Polynomial When Given Zeros Write the basic form of a factored polynomial: Fill in each “c” with a zero Fractions can be written with the denominator in front of the “x” If a complex (imaginary) number is a zero, so is its complex conjugate Multiply the factors together Use the given point to find the value of an

Example 6 (like HW #29-36) Find an nth degree polynomial function with real coefficients with the following conditions: n = 4 Zeros = {-2, -1/2, i} f(1) = 18

You-Try #6 (like HW #29-36) Find an nth degree polynomial function with real coefficients with the following conditions: n = 3 Zeros = 4, 2i f(-1) = -50

Warm-Up: January 12, 2012 Simplify and write in standard form (refer to 2.1 notes if needed)

Homework Questions?

Descarte’s Rule of Signs Let f(x) be a polynomial with real coefficients The number of positive real zeros of f is either equal to the number of sign changes of f(x) or is less than that number by an even integer. If there is only one variation in sign, there is exactly one positive real zero. The number of negative real zeros of f is either equal to the number of sign changes of f(-x) or is less than that number by an even integer. If f(-x) has only one variation in sign, there is exactly one negative real zero.

Example 7 (like HW #43-56) Find all roots of

You-Try #7 (like HW #43-56) Find all zeros of

Assignment Complete one of the following assignments: Page 302 #1-33 Every Other Odd, 43 OR Page 302 #43-55 Odd Chapter 2 Test next week You may use a 3”x5” index card (both sides)