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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Fundamental Theorem of Algebra
EQ: How do I completely solve a polynomial?
MCC9-12.N.CN.9 Know the Fundamental Theorem of Algebra;
show that it is true for quadratic polynomials.
Jan 20­5:04 PM
Warm­up
1. P(x) = 6x4 ­ 162x
2. P(x) = x6 ­ 14x4 + 49x2
3. P(x) = 40 ­ 5x3
Jan 20­5:50 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Rational Roots Theorem
If the polynomial P(x) has integer coefficients, then every rational root of the polynomial equation P(x) = 0 can be written in the form where:
­ p is a factor of the CONSTANT TERM of P(x) ­ q is a factor of the LEADING COEFFICIENT of P(x)
­ must remove GCF before evaluating possible rational roots
Jan 20­5:51 PM
Example 1: Find all possible rational roots P(x) = 8x3 ­ 4x2 ­ 50x + 25
Jan 20­5:50 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Example 2: Find all possible rational roots P(x) = 3x3 ­ 21x2 ­ x + 7
Jan 20­5:50 PM
Example 3: Find all possible rational roots P(x) = x6 ­ 1
Jan 20­5:50 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Example 4: Find all possible rational roots P(x) = 24x2 + 3x5
Jan 20­5:50 PM
Example 5: Find all possible rational roots P(x) = 9x9 ­ 16x7 + 9x6 ­ 16x4
Jan 20­5:50 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Steps for solving a polynomial:
1. List all possible rational roots of a polynomial.
2. Begin testing roots using synthetic division. Remember, the same root might be used multiple times.
3. Keep factoring remaining polynomial until unfactorable or a degree of 2.
4. Set all factors equal to 0 and solve.
Jan 21­4:59 PM
Ex. 1
P(x) = x4 + x3 + 2x2 + 4x ­ 8
Jan 21­5:02 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
Ex. 2
January 22, 2015
P(x) = 2x4 ­ 5x3 ­ 5x2 + 5x + 3
Jan 21­5:02 PM
Ex. 3
P(x) = 2x4 ­ x3 ­ 15x2 + 8x + 20
Jan 21­5:02 PM
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12 ­ Notes ­ Fundamental Theorem of Algebra day 2.notebook
January 22, 2015
Homework:
Rational Roots Worksheet
#1­8 all, #9­19 odd
Jan 22­8:43 AM
Jan 22­8:38 AM
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