Question

9. Find the polynomial of degree 4 whose graph is shown. (It is not necessary to multiply out - You can leave your answer in factored form) 10. Find all real and complex zeros of the function f(x) = 3x* +5x' +25x +45x-18. 11. Find a polynomial of degree 4 that has zeros of 1, 2, and 1+i.
9-11

Answer 1

first remember a point is graph touches Xaxis then zero has o multiplicity and if intersect then have multiplicity! X=-2 Ctouches) Xt2=0 multiplicity should 2 (Oct2)²=0 touches X=1 x-1=0 multiplicit should 2 (x-1)² =0 -0 0 x SC-103c+22=0 focs i COC-120C +232 y intercept = 4 f(0)=4 = K (0-12 (0+272 4 =K4 Ik : f(0) CX-112C+27 2 10] f(Qc) = 3004 +5x3 +2 soch tusa-ig P= roots of -18 Fly 12, 13, 69,718 q=roots of 3 I 1913 P = 51, +2, +3m +6,19, 118,1 ſn +} So now we have to solve it analytically f(-2)=0 OX=-2 using synthetic division 2 SL75 -18 -2 -6 18 3 3003 - Oc2+273-9 5 $4 2 27 CS ned with Camionner

we again have to check analytically so for x = 1/3 wh 3 - that new equation is zero so by synthetic division -1 27 9 9 3 27 o 300² +27 now solving this equation 3x²+27=0 OC - +31 hence zeros are OC = = -2, +31,-3;3 !1] zero are lo 22 Iti forth root willle Hi because complex roots aluay comes in conjunta O: 1 x-1=0 - x=2 X-2-0 x=lti X-Citi) -3 Q=1-1 3-(1-1) -3 multiply 1, 2, © E (OC-1) COC-2) Ć 2-7-1) ca-Iti) 9 foc) = Coc? +2 -330) C002 -20C +1 +1) = cc?+2 -30C) (oc² -2x+2) = och 5x² #40c2 tox + 4 - 04-50c 3 + 100100c+4 USIV Sollow f(-2)=0 -S4 18 -2 oX=-2 3 5 2 s ur S -18 -6 2 -1 27 3003 - C2+273-9 CS Scanned with Cancer