Polynomial FunctionsENGR CJM PINEDALong Division and Synthetic DivisionExample 1Use the long division method to divide byx!"$ol%tionDivide leading te&ms'M%ltiply' $%bt&a(t and b&ing do)n 1*+hen &epeat the p&o(ess %sing the last line as dividendDivide leading te&ms'M%ltiply' $%bt&a(t and b&ing do)n !DividendDiviso&,%otientRemainde&$olve %sing syntheti( division$ol%tion )&ite only the (oe-(ientb&ing do)n .m%ltiply' /.0/!01*!add 2*.3*!12*$olve %sing syntheti( division$ol%tion 4%otient&emainde&Long Division and Synthetic DivisionExample *Use syntheti( division to divide by x 3 *"Example 5Divide' Remainder TheoremIn example *6 /x 3 *0 has no &emainde&" 7hat does this imply8/x 3 *0 is a 9a(to& o9 Conside& the 9%n(tion Is the&e a &emainde& )hen )e divide P/x0 by d/x08+his sho)s that /x 3 50 is not a 9a(to& o9 P/x0"Eval%ate P/250"7hat do yo% obse&ve8Remainder and Factor TheoremsAs a (onse4%en(e o9 the Remainde& +heo&em6 )hen P/x0 is divided by /x(0 and the &emainde& is :6 P/(0 1 :6 and ( is a ;e&o o9 the polynomial" +h%s6 )e have the 0 9o& Example >Use the 9a(to& theo&em to dete&mine i9 /a0 x *6 /b0 x 3 1 a&e 9a(to&s o9 Finding Zeros of a PolynomialExample .Given that * is a ;e&o6 %se the 9a(to& theo&em to help =nd all othe& ;e&os"Zeros of a Polynomial+he zeros of a polynomial 9/x0 a&e the sol%tions o9 the e4%ation 9/x0 1 :" Ea(h &eal ;e&o is an x2inte&(ept o9 the g&aph o9 f"Linear Factorization TheoremIn othe& )o&ds6 eve&y (omplex polynomial o9 deg&ee n (an be &e)&itten as the p&od%(t o9 a non;e&o (onstant and exa(tly n linea& 9a(to&s"Linear Factorization TheoremExample ?Re)&iteas a p&od%(t o9 linea& 9a(to&s6 and =nd its ;e&os"Example @