1. Today in Pre-Calculus Go over homework Notes: –Real Zeros of polynomial functions –Rational Zeros Theorem Homework

2. Real Zeros of Polynomial Functions Real zeros of polynomial functions are either: Rational zeros: f(x) = x 2 – 16 (x – 4)(x + 4) = 0 x = 4, -4 OrIrrational zeros: f(x) = x 2 – 3

3. Rational Zeros Theorem tells us how to make a list of potential rational zeros for a polynomial function with integer coefficients. If p be all the integer factors of the constant and q be all the integer factors of the leading coefficient in the polynomial function then, gives us a list of potential rational zeros

4. Example Use the rational zeros theorem to find the potential zeros of f(x) = 2x 4 + 5x 3 – 13x 2 – 22x + 24 then find all zeros

5. Example (cont.) 2 2 5 -13 -22 24 2 4 95 1810 -12 0 f(x) = (x – 2)(x 3 + 2x 2 – 3x – 6) This proves 2 is a zero

6. Example (cont.) f(x)= (x – 2)(x + 3)(2x 2 +3x – 4) The remaining factor is quadratic, so factor or use the quadratic formula. 2 2 9 5 -12 -3 -6 3-4 -912 0

7. Example Use the rational zeros theorem to find the potential zeros of f(x) = x 4 + 4x 3 – 7x 2 – 8x + 10 then find all zeros

8. Example (cont.) 1 1 4 -7 -8 10 -5 -2 510 2 -10 0 f(x) = (x + 5)(x 3 - x 2 – 2x + 2)

9. Example (cont.) f(x)= (x – 1)(x + 5)(x 2 – 2) 1 1 -1 -2 2 1 1 0-2 0 0

10. Homework Pg. 224: 34-36, 50, 54 Quiz: Monday

11. Upper and Lower Bound Tests for Real Zeros Helps to narrow our search for all real (rational and irrational) zeros Helps to know that we have found all the real zeros since a polynomial can have fewer zeros than its degree. (Remember a polynomial with degree n has at most n zeros.) Upper Bound: k is an upper bound if k > 0 and when x – k is synthetically divided into the polynomial, the values in the last line are all non-negative. This means all of the real zeros are smaller than or equal to k. Lower Bound: k is a lower bound if k < 0 and when x – k is synthetically divided into the polynomial, the values in the last line are alternating non-positive and non-negative. This means all of the real zeros are greater than or equal to k.

12. Example If f(x) = x 4 – 7x 2 + 12 prove that all zeros are in the interval [-4, 3]. 1 1 0 -7 0 12 3 3 32 96 6 18 30 1 1 0 -7 0 12 -4 9 16-36 144 156 All are non- negative, So 3 is the upper bound Alternate between non-positive and non-negative, so -4 is the lower bound.