Today in Pre-Calculus

• Go over homework• Notes:

– Real Zeros of polynomial functions

– Rational Zeros Theorem• Homework

Real Zeros of Polynomial FunctionsReal 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 3 0x x

3, 3x

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 zerosp

q

Example

Use the rational zeros theorem to find the potential zeros of f(x) = 2x4 + 5x3 – 13x2 – 22x + 24 then find all zeros

1, 2, 3, 4, 6, 8, 12, 24

1, 2

1 31, 2, 3, 4, 6, 8, 12, 24, ,

2 2

Example (cont.)

2

2 5 -13 -22 242

4

9 5

18 10

-12

-12

0

f(x) = (x – 2)(x3 + 2x2 – 3x – 6)

This proves 2 is a zero

Example (cont.)

f(x)= (x – 2)(x + 3)(2x2 +3x – 4)

The remaining factor is quadratic, so factor or use the quadratic formula.

3 41 3 41,

4 4x

2

2 9 5 -12-3

-6

3 -4

-9 12

0

3 41 3 41List of zeros: 2, 3, ,

4 4x

Example

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

1, 2, 5, 101, 2, 5, 10

1

Example (cont.)

1

1 4 -7 -8 10-5

-5

-1 -2

5 10

2

-10

0

f(x) = (x + 5)(x3 - x2 – 2x + 2)

Example (cont.)

f(x)= (x – 1)(x + 5)(x2 – 2)

2x

1

1 -1 -2 21

1

0 -2

0 -2

0

List of zeros: 1, 5, 2x

Homework

• Pg. 224: 34-36, 50, 54

• Quiz: Monday

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.

ExampleIf f(x) = x4 – 7x2 + 12 prove that all zeros are in the interval [-4, 3].

1

1 0 -7 0 123

3

3 2

9 6

6

18

30

1

1 0 -7 0 12-4

-4

-4 9

16 -36

-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.