1. ROOTS OF POLYNOMIALS DUBAN CASTRO FLOREZ PETROLEUM ENGINEERING NUMERICS METHODS IN ENGINEERING 2010 1
2. POLYNOMIAL A polynomial is a sum of terms called monomials. A monomial is the product of a coefficient (a real number), a variable (almost always x or y) high to an exponent (I inform positive). Examples: is a degree polynomial 7. is a degree polynomial 5. 2
3. ROOTS OF A POLYNOMIAL The root of a polynomial is a such number that makes that the polynomial is worth zero. That is to say that, when we solve a polynomial to zero, the solutions are the roots of the polynomial. For example the polynomial f(x) = x2 + x - 12 When we equal it to zero and we solve it we have: x2 + x - 12 = 0 Equaling to zero. (x + 4)(x - 3) = 0 Factoring. x = - 4 Solution 1 x = 3 Solution 2 3
4. 4 Since x1 = - 4 and x2 = 3 are f(x solutions) then f (-4) = 0 and f (3) = 0. We say then that x = - 4 and x = 3 are roots of the polynomial f(x) = x2 + x - 12
5. FUNDAMENTAL THEOREM OF THE ALGEBRA Carl Friedrich Gauss has been one of the biggest mathematicians in all the times. It contributed to many branches of the mathematics. In 1798, Gauss demonstrated the fundamental theorem of the Algebra that it says the following thing:” ALL DEGREE POLYNOMIAL n HAS n ROOTS” That is to say that the equation anxn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + ... + a3 x3 + a2 x2 + a1 x + a0 = 0 it has n solutions. 5
6. DESCARTES RULE Rene Descartes (the same of the Cartesian plane) found a method to indicate the number of positive roots in a polynomial. This rule says the following thing: "The number of positive real roots of a polynomial f(x) it is similar to the number of changes of term sign to f(x )“ For example the polynomial : f(x) = x2 + x - 12 have a sign change, of the second to the third term, therefore it has a positive root. g(x) = +x3 - 4 x2 + x + 6 have two sign changes, it has two positive roots h(x) = +x4 - 5 x2 + 4 have two positive roots 6
7. PROPERTIES OF THE ROOTS AND FACTORS OF A POLYNOMIAL 1. The zeros or roots are dividing of the independent term of the polynomial. 2. To each root of the type x = a it corresponds him a binomial of the type (x-a). 3. Can express a polynomial in factors when writing it as product of all the binomials of the type (x - a) that belong together to the roots, x = a that are obtained. x2− 5x + 6 = (x − 2) · (x − 3) 7
8. PROPERTIES OF THE ROOTS AND FACTORS OF A POLYNOMIAL 4. The sum of the exponents of the binomials must be similar to the degree of the polynomial. 5. all polynomial that doesn't have independent term admits as root x = 0, or what is the same thing, admits as factor x. x2 + x = x · (x + 1) Roots: x = 0 y x = − 1 6. A polynomial is called irreducible or I prevail when it cannot break down in factors. P(x) = x2 + x + 1 8
9. EXAMPLES To find the roots of the following polynomials: 1.f(x)= x3 - 4 x2 + x + 6 2.f(x)= x4 - 5 x2 + 4 3. f(x)= x2 - 6x + 5 9
