Let f(x)=ax4+bx3+cx2+dx+e, then we have a system of equations
⎩⎨⎧f(−1)=a−b+c−d+e=3f(3)=81a+27b+9c+3d+e=39f(6)=1296a+216b+36c+6d+e=822f(7)=2401a+343b+49c+7d+e=1611f(0)=e=−6
Extended matrix of the system is
⎝⎛181129624010−12721634301936490−13670111113398221611−6⎠⎞
Solve the system by Jordan method
⎝⎛181129624010−12721634301936490−13670111113398221611−6⎠⎞→
→⎝⎛181129624010−12721634301936490−13670000019458281617−6⎠⎞→
→⎝⎛10000−11081512274401−72−1260−23520−184130224080000019−684−10836−19992−6⎠⎞→
→⎝⎛10000−19364901−6−30−420−1731430000019−57−258−357−6⎠⎞→
→⎝⎛100000900031−6−6−3280−927394400000138−57−30−3140−6⎠⎞→
→⎝⎛100000900031−6−2−840−92714400000138−57−10−420−6⎠⎞→
→⎝⎛100000900000−200−1814120000011−27−100−6⎠⎞→
→⎝⎛100000900000−20000010000011−27−100−6⎠⎞→
→⎝⎛10000010000010000010000011−350−6⎠⎞
We obtain a=1,b=−3,c=5,d=0,e=−6, that is f(x)=x4−3x3+5x2−6
Answer: f(x)=x4−3x3+5x2−6
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