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LESSON 14 –
FUNDAMENTAL THEOREM
of ALGEBRA
PreCalculus - Santowski
(A) Opening Exercises
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Solve the following polynomial equations where x  R
Simplify all solutions as much as possible
Rewrite the polynomial in factored form
PreCalculus - Santowski
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x3 – 2x2 + 9x = 18
x3 + x2 = – 4 – 4x
4x + x3 = 2 + 3x2
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(A) Opening Exercises
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Solve the following polynomial equations where x  C
Simplify all solutions as much as possible
Rewrite the polynomial in factored form
PreCalculus - Santowski
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5/24/2017
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x3 – 2x2 + 9x = 18
x3 + x2 = – 4 – 4x
4x + x3 = 2 + 3x2
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LESSON OBJECTIVES
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PreCalculus - Santowski
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State and work with the Fundamental Theorem
of Algebra
Find and classify all real and complex roots of a
polynomial equation
Write equations given information about the
roots
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(A) FUNDAMENTAL THEOREM OF
ALGEBRA
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So far, in factoring higher degree polynomials, we
have come up with linear factors and irreducible
quadratic factors when working with real
numbers
PreCalculus - Santowski
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But when we expanded our number system to
include complex numbers, we could now factor
irreducible quadratic factors
So now, how many factors does a polynomial
really have?
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(A) FUNDAMENTAL THEOREM OF
ALGEBRA
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The fundamental theorem of algebra is a statement about
equation solving
PreCalculus - Santowski
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5/24/2017
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There are many forms of the statement of the FTA  we
will state it as:
If p(x) is a polynomial of degree n, where n > 0, then f(x)
has at least one zero in the complex number system
A more “useable” form of the FTA says that a polynomial of
degree n has n roots, but we may have to use complex
numbers.
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(A) FUNDAMENTAL THEOREM OF
ALGEBRA
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A more “useable” form of the FTA says that a polynomial of
degree n has n roots, but we may have to use complex
numbers.
PreCalculus - Santowski
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So what does this REALLY mean for us given cubics &
quartics?  all cubics will have 3 roots and thus 3 linear
factors and all quartics will have 4 roots and thus 4 linear
factors
So we can factor ANY cubic & quartic into linear factors
And we can write polynomial equations, given the roots of
the polynomial
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(B) WORKING WITH THE FTA
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PreCalculus - Santowski
Solve the following polynomials, given that xεC.
Round all final answers to 2 decimal places
where necessary.
(i) x3 – 8x2 + 25x – 26 = 0
 (ii) x3 + 13x2 + 57x + 85 = 0
 (iii) x3 – 4x2 + 4x – 16 = 0
 (iv) x3 – 10x2 + 34x – 40 = 0
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(B) WORKING WITH THE FTA
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PreCalculus - Santowski
Solve the following polynomials, given that xεC.
Round all final answers to 2 decimal places
where necessary.
(i) x4 – 7x3 + 19x2 – 23x + 10 = 0
 (ii) x4 – 3x2 = 4
 (iii) 2x4 + x3 + 7x2 + 4x – 4 = 0
 (iv) x4 + 2x3 – 3x2 = -6 – 2x
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(C) WORKING WITH THE FTA – GIVEN ROOTS
(i) one root of x3 + 3x2 + x + 3 is i
 (ii) one root of 2x3 – 17x2 + 42x – 17 is ½
 (iii) one root of x4 – 5x3 – 3x2 + 43x – 60 is 2 + i
 (iv) one root of 4x4 – 4x3 – 15x2 + 38x – 30 is 1 - i
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PreCalculus - Santowski
In this question, you are given information about
some of the roots, from which you can find the
remaining zeroes, write the factors, from which
you can write the equation in standard form
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(C) WORKING WITH THE FTA – GIVEN ROOTS
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PreCalculus - Santowski
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For the following polynomial functions x  C
State the other complex root
Rewrite the polynomial in factored form
Expand and write in standard form
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(i) one root is -2i as well as -3
(ii) one root is 1 – 2i as well as -1
(iii) one root is –i, another is 1-i
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(C) WORKING WITH THE FTA – GIVEN ROOTS
(i) the roots of a cubic are -2 and i
 (ii) the roots of a quartic are 3 (with a multiplicity
of 2) and 1 – i
 (iii) the roots of a cubic are -1 and 2 + 3i
 (iv) the roots of a quartic are 2i and 3 - i
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PreCalculus - Santowski
In this question, you are given information about
the roots, from which you can find the remaining
zeroes, write the factors, from which you can
write the equation in standard form
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(D) WORKING WITH THE FTA
PreCalculus - Santowski
Given a graph of p(x),
determine all roots and
factors of p(x)
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Given a graph of p(x),
determine all roots and
factors of p(x)
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