Today in Pre-Calculus

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) = x2 – 16 (x – 4)(x + 4) = 0 x = 4, -4 Or Irrational zeros: f(x) = x2 – 3

3. Example Find all the real zeros of f(x) = 2x4 + 5x3 – 13x2 – 22x + 24. Answers must be exact, not decimal approximations. 2 5 -13 -22 24 2 18 10 -12 4 2 9 5 -12 0 This proves 2 is a zero f(x) = (x – 2)(x3 + 2x2 – 3x – 6)

4. Example (cont.) 2 9 5 -12 -3 -9 12 -6 f(x)= (x – 2)(x + 3)(2x2 +3x – 4) The remaining factor is quadratic, so factor or use the quadratic formula. 2 3 -4 0

5. Example (cont.) Find all of the real zeros of f(x) = x4 + 4x3 – 7x2 – 8x + 10 1 4 -7 -8 10 -5 5 10 -10 -5 1 -1 -2 2 0 f(x) = (x + 5)(x3 - x2 – 2x + 2)

6. Example (cont.) 1 -1 -2 2 1 0 -2 1 f(x)= (x – 1)(x + 5)(x2 – 2) 1 0 -2 0

7. Homework • Pg. 225:, 50, 51, 54 • Quiz:

8. 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 mostn 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 alternatingnon-positive and non-negative. This means all of the real zeros are greater than or equal to k.

9. Example If f(x) = x4 – 7x2 + 12 prove that all zeros are in the interval [-4, 3]. 1 0 -7 0 12 All are non-negative, So 3 is the upper bound 3 9 6 18 3 1 3 2 30 6 1 0 -7 0 12 -4 16 -36 144 -4 1 -4 9 156 -36 Alternate between non-positive and non-negative, so -4 is the lower bound.

10. 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

11. Example Find the potential zeros of f(x) = 2x4 + 5x3 – 13x2 – 22x + 24 then find all zeros

12. Example Use the rational zeros theorem to find the potential zeros of f(x) = x4 + 4x3 – 7x2 – 8x + 10 then find all zeros