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(A) Opening Exercises Solve the following polynomial equations where Simplify all solutions as much as possible Rewrite the polynomial in factored form x 3 – 2x 2 + 9x = 18 x 3 + x 2 = – 4 – 4x 4x + x 3 = 2 + 3x 2 2/16/ PreCalculus - Santowski
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LESSON 14 – FUNDAMENTAL THEOREM of ALGEBRAPreCalculus - Santowski
(A) Opening Exercises
Solve the following polynomial equations where
Simplify all solutions as much as possible Rewrite the polynomial in factored form
x3 – 2x2 + 9x = 18 x3 + x2 = – 4 – 4x 4x + x3 = 2 + 3x2
05/04/23
2
PreC
alculus - Santow
ski
x R
(A) Opening Exercises
Solve the following polynomial equations where
Simplify all solutions as much as possible Rewrite the polynomial in factored form
x3 – 2x2 + 9x = 18 x3 + x2 = – 4 – 4x 4x + x3 = 2 + 3x2
05/04/23
3
PreC
alculus - Santow
ski
Cx
LESSON OBJECTIVES
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
05/04/23
4
PreC
alculus - Santow
ski
(A) FUNDAMENTAL THEOREM OF ALGEBRA So far, in factoring higher degree
polynomials, we have come up with linear factors and irreducible quadratic factors when working with real numbers
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?
05/04/23
5
PreC
alculus - Santow
ski
(A) FUNDAMENTAL THEOREM OF ALGEBRA The fundamental theorem of algebra is a statement
about equation solving
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.
05/04/23
6
PreC
alculus - Santow
ski
(A) FUNDAMENTAL THEOREM OF ALGEBRA 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.
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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PreC
alculus - Santow
ski
(B) WORKING WITH THE FTA 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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PreC
alculus - Santow
ski
(B) WORKING WITH THE FTA 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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PreC
alculus - Santow
ski
(C) WORKING WITH THE FTA – GIVEN ROOTS 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
(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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PreC
alculus - Santow
ski
(C) WORKING WITH THE FTA – GIVEN ROOTS
For the following polynomial functions State the other complex root Rewrite the polynomial in factored form Expand and write in standard form
(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
05/04/23
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PreC
alculus - Santow
ski
Cx
(C) WORKING WITH THE FTA – GIVEN ROOTS 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
(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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PreC
alculus - Santow
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(D) WORKING WITH THE FTA
Given a graph of p(x), determine all roots and factors of p(x)
Given a graph of p(x), determine all roots and factors of p(x)
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reCalculus - S
antowski
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