LESSON 14 – FUNDAMENTAL THEOREM of ALGEBRA PreCalculus - Santowski

(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

(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

L ESSON O BJECTIVES 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 2/16/ PreCalculus - Santowski

(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? 2/16/ PreCalculus - Santowski

(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. 2/16/ PreCalculus - Santowski

(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 2/16/ PreCalculus - Santowski

(B) W ORKING WITH THE FTA Solve the following polynomials, given that xε C. Round all final answers to 2 decimal places where necessary. (i) x 3 – 8x x – 26 = 0 (ii) x x x + 85 = 0 (iii) x 3 – 4x 2 + 4x – 16 = 0 (iv) x 3 – 10x x – 40 = 0 2/16/ PreCalculus - Santowski

(B) W ORKING WITH THE FTA Solve the following polynomials, given that xε C. Round all final answers to 2 decimal places where necessary. (i) x 4 – 7x x 2 – 23x + 10 = 0 (ii) x 4 – 3x 2 = 4 (iii) 2x 4 + x 3 + 7x 2 + 4x – 4 = 0 (iv) x 4 + 2x 3 – 3x 2 = -6 – 2x 2/16/ PreCalculus - Santowski

(C) W ORKING WITH THE FTA – G IVEN 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 x 3 + 3x 2 + x + 3 is i (ii) one root of 2x 3 – 17x x – 17 is ½ (iii) one root of x 4 – 5x 3 – 3x x – 60 is 2 + i (iv) one root of 4x 4 – 4x 3 – 15x x – 30 is 1 - i 2/16/ PreCalculus - Santowski

(C) W ORKING WITH THE FTA – G IVEN 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 -2 i as well as -3 (ii) one root is 1 – 2 i as well as -1 (iii) one root is – i, another is 1- i 2/16/ PreCalculus - Santowski

(C) W ORKING WITH THE FTA – G IVEN 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 i (iv) the roots of a quartic are 2 i and 3 - i 2/16/ PreCalculus - Santowski

(D) W ORKING WITH THE FTA Given a graph of p(x), determine all roots and factors of p(x) 2/16/2016 PreCalculus - Santowski 13