Warm – up #51. Find all the zeros of P(x), given is a zero,

Deg = 4 4 roots1+𝑖 −2 1 2

−1+𝑖−2−1

−1−𝑖1−𝑖

1

11+𝑖 −2201−𝑖

00−1

−1+𝑖01

1−𝑖𝑥¿𝑅𝑥2

Warm – up #51. Solve Solve by square root, factoring, or quadratic formula(x – 1)(x + 1) = 0x – 1 = 0x + 1 = 0x = 1 x = –1Or

Homework LogFri

1/8

Lesson 5 – 4

Learning Objective: To apply Rational Zeros Theorem

Hw: #507 Pg. 302 #1 – 19 odd

1/8/16 Lesson 5 – 4 Rational Zeros Day 1

Advanced Math/Trig

Learning Objective To apply the Rational Zeros Theorem

To apply Bound Theorem

Rational Zeros Theorem

Rational #: Quotient of integers , If is a zero of a polynomial with integer coefficients, then p is a factor of the constant term, and q is a factor of the leading coefficient

List all possible rational zeros++…+

Rational Zeros Theorem

q = factors of p = factors of

Possible Rational Zeros = combos

1.

Give the set of possible rational zeros

of P(x)

𝑞=±1 , ±3 𝑝=±1 , ±5 =

, ,

2.

Give the set of possible rational zeros

of P(x)

𝑞=±1𝑝=±1 , ±2 , ±3 ,±6 =

Bounds TheoremUsing synthetic division on poly with real coeff & lead coeff is positive:

(1) If c is a positive real, & all #s in last row are non-negative, c is an upper bound

(no zeros > c)(2) If c is a negative real, & the #s

in last row alternate in sign, c is a lower bound

(no zeros < c)

(0 can be written as +0 or -0)

a) Find smallest possible integer that bounds theorem detects as an upper

bound for the zeros of

13 – 7 15 – 35

32

– 43 – 1

3 3 2 21 28

Stop when neg.

All non-neg.3 is an upper bound

3.

b) Find the negative integer nearest zero that bounds theorem detects as a

lower bound for the zeros of

– 13 – 7 15 – 35

3 – 10 25 – 60 Alt. signs

–1 is a lower bound

3.

Find upper & lower bounds for the zeros

of

11 – 1 – 4 – 2 – 13

1 0 – 42 1 1 – 23 1 2 2 4 – 14 1 3 8 30 107– 1 1 – 2 – 2– 2 1 – 3 2– 3 1 – 4 8 – 26 65

– 6 – 1

4 is upperbound

–3 is lowerbound

Stop when neg.

Stop when not alternating

4.

Find upper & lower bounds for the zeros

of

11 1 – 11 1 – 10

1 2 – 92 1 3 – 53 1 4 1 4 2– 1 1 0 –12– 2 1 – 1 – 9– 3 1 – 2 –5– 4 1 – 3 1 – 3 2

3 is upperbound

– 4 is lowerbound

Stop when neg.

Stop when not alternating

5.

Ticket Out the Door Find upper & lower bounds for the zeros of

Homework#507 Pg. 302 #1 – 19 odd