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

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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

x3 – 2x2 + 9x = 18 x3 + x2 = – 4 – 4x 4x + x3 = 2 + 3x2

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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

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(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?

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(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.

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(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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(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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(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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(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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(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

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(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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(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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