0 wed 12/2 lesson 5 – 5 learning objective: to apply rational root thm (& descartes’rule)...
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1.
Warm – up #4 Synthetic Div
x = 𝑥3 𝑥2 𝑥¿−1 −2 −3 0
−2
2
0
0
−33
3𝑥2 𝑥¿ 𝑅
𝑥4
1
1𝑥3
−1
=
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Homework Log
Wed
12/2
Lesson 5 – 5
Learning Objective: To apply Rational Root Thm (& Descartes’Rule)
Hw: W.S. 5-5
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12/2/15 Lesson 5 – 5 Rational Root Theorem
Day 1Algebra II
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To understand & apply the Rational Root Theorem
To understand & apply the Conjugate Root Theorem
(To understand and apply Descartes’ Rule of Signs)
Learning Objective
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List all possible rational zeros++…+
Rational Root Theorem
q = factors of p = factors of
Possible Rational Zeros = combos
Look at COEFFICIENTS
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1. List all possible rational zeros
Rational Root Theorem
q = 1 p = 12
=
= 2, 3, 4, 6, 12
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2. List all possible rational zeros
Rational Root Theorem
q = 3 p = 10
=
= 2, , , , ,
Divide each numerator by each denominator
112151
101
1323
53
103
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3. What are the rational roots of
Applying Rational Root Theorem
q = 1 p = 42
Step 1: List all the possible rational zeros
=
= 2, , , , ,
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3. What are the rational roots of
Applying Rational Root Theorem
2, , , , ,
Step 2: Test each possible rational root in original function until you find a root.
Do synthetic substitution until get R0
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3.
Applying Rational Root Theorem
1𝑥3 𝑥2 𝑥 ¿1 −6 −13 42
11−5−5−18
−1824𝑅
-11 −6 −13 42
1−1−77−6648𝑅
Nope! 1 is NOT a root!
Nope! 1 is NOT a root!
Try something else!
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3.
Applying Rational Root Theorem
2𝑥3 𝑥2 𝑥 ¿1 −6 −13 42
12−4−8−21
−420 YES! 2 IS a
root!
Factor & Solve to find rest of the roots! (x-2)(x – 7)(x + 3)
x = 2, 7, –3
𝑥2 𝑥¿ 𝑅
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4. What are the rational roots of
Applying Rational Root Theorem
q = 2 p = 6
=
= 2, , , ,
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4.
Applying Rational Root Theorem
-1𝑥3 𝑥2 𝑥 ¿21 −7 −6
2−2−11−660 YES! -1 IS a root!
We have (x+1)( Factor & Solve to find rest of the roots! (x+1)(2x + 3)(x - 2)
x = -1,, 2
𝑥2 𝑥¿ 𝑅
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If is a root of P(x), then is also a root of P(x)
If is a root of P(x), then is also a root of P(x)
Conjugate Root Theorem
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5. and are roots of P(x), find two other roots
and
6. and are roots of P(x), find one other root.
Conjugate Root Theorem
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Stop here! Do rest tomorrow!
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The biggest exponent of tells you how many roots there are
The # of times changes signs tells you how many possible (+) roots there are or less that by an even #
The # of times changes signs tells you how many possible (-) roots there are or less that by an even #
Descartes’ Rule of Signs
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7.
Descartes’ Rule of Signs
2 Sign Change
1 Sign Change+ - 2 1 00 1 2
, so should have 3 solutions total
Sum = 3
Sum = 3
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8.
Descartes’ Rule of Signs
5 Sign Change
4 Sign Change
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8. , so should have 9 solutions totalDescartes’ Rule of Signs
+ -
5 4 05 2 25 0 43 4 23 2 43 0 61 4 41 2 61 0 8
It could have 5 positive roots, 4 negative roots, & 0 imaginary roots
It could have 3 positive roots, 2 negative roots, & 4 imaginary roots
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9.
Descartes’ Rule of Signs
3 Sign Change
1 Sign Change+ - 3 1 01 1 2
It could have 3 + roots, 1 - root, & 0 imaginary roots
It could have 1 + roots, 1 - root, & 2 imaginary roots
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10. What are the rational roots of
Applying Rational Root Theorem
q = 1 p = 6
=
= 2, ,
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10.
Applying Rational Root Theorem
1𝑥3 𝑥2 𝑥 ¿1 −1 6 −6
1100660 YES! 1 IS a root!
is not factorable, so the only RATIONAL ROOT is 1
*Can solve = 0 to find irrational roots
𝑥2 𝑥¿ 𝑅
x =1,
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What does Descartes’ Rule of Signs say about the # of positive, negative, & imaginary roots of
Ticket Out the Door
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Assignment:
W.S. 5-5