Find the sum of the coefficients of the polynomial P(x)=x4−29x3+ax2+bx+c, given that P(5)=11,P(11)=17, and P(17)=23.
A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.
Solution
Define Q(x)=P(x)−x−6=x4−29x3+ax2+(b−1)x+(c−6) and notice that Q(5)=Q(11)=Q(17)=0. Q(x) has degree 4 and by Vieta's Formulas the sum of its roots is 29, so its last root is 29−17−11−5=−4, giving us Q(x)=(x−5)(x−11)(x−17)(x+4). This means that P(1)=Q(1)+7=(−4)(−10)(−16)(5)+7=−3200+7=−3193.
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