Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 19638 by Tinkutara last updated on 13/Aug/17

$$\mathrm{Let}P\left(x\right)\mathrm{is}\mathrm{a}\mathrm{polynomial}\mathrm{such}\mathrm{that}$$$$P\left(1\right)=1,P\left(2\right)=2,P\left(3\right)=3,\mathrm{and}$$$$P\left(4\right)=5.\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}P\left(6\right).$$

Answered by ajfour last updated on 13/Aug/17

$$\mathrm{let}\mathrm{P}\left(\mathrm{x}\right)={\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}$$$$\mathrm{P}\left(1\right)=1,\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=1$$$$\mathrm{P}\left(2\right)=2,\Rightarrow 8\mathrm{a}+4\mathrm{b}+2\mathrm{c}+\mathrm{d}=2$$$$\mathrm{P}\left(3\right)=3,\Rightarrow 27\mathrm{a}+9\mathrm{b}+3\mathrm{c}+\mathrm{d}=3$$$$\mathrm{P}\left(4\right)=5,\Rightarrow 64\mathrm{a}+16\mathrm{b}+4\mathrm{c}+\mathrm{d}=5$$$$\Rightarrow 37\mathrm{a}+7\mathrm{b}+\mathrm{c}=2$$$$19\mathrm{a}+5\mathrm{b}+\mathrm{c}=1$$$$7\mathrm{a}+3\mathrm{b}+\mathrm{c}=1$$$$\Rightarrow 18\mathrm{a}+2\mathrm{b}=1$$$$12\mathrm{a}+2\mathrm{b}=0$$$$\Rightarrow 6\mathrm{a}=1\mathrm{or}\mathrm{a}=\frac{1}{6}$$$$\mathrm{b}=-1,\mathrm{c}=1-3\mathrm{b}-7\mathrm{a}$$$$\Rightarrow \mathrm{c}=1+3-\frac{7}{6}=\frac{17}{6}$$$$\mathrm{d}=1-\mathrm{a}-\mathrm{b}-\mathrm{c}$$$$=1-\frac{1}{6}+1-\frac{17}{6}=-1$$$$\mathrm{P}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^3}{6}-{\mathrm{x}}^2+\frac{17\mathrm{x}}{6}-1$$$$\mathrm{P}\left(6\right)=36-36+17-1=16.$$

Commented by Tinkutara last updated on 13/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{Sir}!$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com