Wed 12/9 Lesson Rev Learning Objective: To remember everything about Polynomials! Hw: Quiz Corrections.
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Homework Log
Wed
12/9
Lesson Rev
Learning Objective: To remember everything about Polynomials!
Hw: Quiz Corrections
Homework Log
Wed
12/9
Lesson Rev
Learning Objective: To remember everything about Polynomials!
Hw: Test Review WS
12/9/15 Chapter 5 Review
Algebra II
To remember everything about Chapter 5
Learning Objective
1. Quintic Binomial
Write in standard form & classify by degrees & number of terms.
2. What is a cubic polynomial function in standard form with zeros –2, , and 0?x = –2 x = x = 0x + 2 = 0 3x = 4 x = 0x + 2 = 0 3x - 4 =0 x = 0(x + 2)(3x – 4)(x) = 0 (
Write a Polynomial Function From its Zeros
3. Write a polynomial function with rational coefficients so that
P(x) = 0 has the given roots of and
must also be a root!x = x = x = x + 3 = 0 x – = 0 x + = 0
(x + 3)(x - )(x + ) = 0
Write a Polynomial Function From its Roots
(x + 3)(x - )(x + ) = 0
- x
–x 𝑥2
-49x
x
7 𝑖
3 x
3 𝑥3
14749x
49
-49 = 49
4. Write a polynomial function with rational coefficients so that
P(x) = 0 has the given roots of
must also be a root!x = x = x=0 x=0
(x)(x) = 0
Write a Polynomial Function From its Roots
(x)(x) = 0
-3x𝑥2
9-3x
x
-3
-3𝑥 2√5
−6 √5 6 √5−4 √25
= 0
= -20
5. (2x – 3)(42x – 3 = 0 4x =
Find all the roots
2x 3
6. Divide by Long Division
4 𝑥−1 8 𝑥3+18 𝑥2+7 𝑥−32 𝑥2
8 𝑥3 -
12x
+5x
20 +7x-
20 -5x
-
--
-312x - 3-
-
0R
+ 3
7.
Divide using Synthetic Division
=
x = 1
1𝑥2 𝑥¿00 −15
1
1
11−14
𝑥¿ 𝑅
𝑥3
1
1𝑥2
1
8. , what is ?Remainder Theorem
-2𝑥3𝑥2 𝑥¿1−5 0 7
17−34−397878
−156−149
3
3−6
𝑅
𝑥4
−2
−8
𝑥5
16
-149
9. Is a factor of ? If it is, find the remaining factors.
Factor Theorem
2𝑥3 𝑥2 𝑥¿10 −1216
1
2
24−8−16
0𝑥2 𝑥¿ 𝑅
R 0, so x - 2 is a factor of P(x)!
+ 2x - 8 Factor more!
9. =
= (x - 2)(x - 2)(x + 4) =
Factor Theorem
10. What are the rational roots of
Applying Rational Root Theorem
, ,
Step 2: Test each possible rational root in original function until you find a root.
Do synthetic substitution until get R0
10.
Applying Rational Root Theorem
1𝑥3 𝑥2 𝑥 ¿11 −17 15
1122−15
−150 YES! 1 IS a
root!Factor & Solve to find rest of the roots!(x – 3)(x + 5)
x = 1, –5, 3
𝑥2 𝑥¿ 𝑅
11. Find all Roots
1𝑥3 𝑥2 𝑥 ¿1 −1 4 −4
1100440 YES! 1 IS a root!
is not factorable, so the only RATIONAL ROOT is 1
*Can solve = 0 to find imaginary roots
𝑥2 𝑥¿ 𝑅
x =1,
,
12.
Descartes’ Rule of Signs
3 Sign Change
2 Sign Change
12. , so should have 5 solutions totalDescartes’ Rule of Signs
+ -
3 2 03 0 21 2 21 0 4
13. What are ALL the roots of
Applying FTA
q = 1 p = 24
=
= 2, 3, 4, 6, 8, 12, 24
13.Applying FTA
2𝑥3 𝑥2 𝑥¿1 −2 4 −24
364812240
𝑥2 𝑥¿ 𝑅
𝑥41
1𝑥3
2
-313 4 12
1−3004
−120
𝑥2 𝑥¿ 𝑅
13.
x =
Applying FTA
Assignment:
Chapter 5 Test Review WS
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