Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 1 - Systems of Linear Equations - 1.3 Applications of Systems of Linear Equations - 1.3 Exercises - Page 32: 6


$$ p(x)=42-41x-2x^2+ x^3. $$

Work Step by Step

Suppose that $$ p(x)=a_{0}+a_{1} x+a_{2} x^{2}+a_3x^3. $$ Now, we have $$ \begin{array}{l} {p(0)=a_{0}+a_{1}(0)+a_{2}(0)^{2}+a_3(0)^3=a_{0}=42} \\ {p(1)=a_{0}+a_{1}(1)+a_{2}(1)^{2}+a_3(01)^3=a_{0}+a_1+a_2+a_3 =0} \\ {p(2)=a_{0}+a_{1}(2)+a_{2}(2)^{2}+a_3(2)^3=a_{0}+ 2a_{1}+ 4a_{2}+8a_3=-40}\\ {p(3)=a_{0}+a_{1}(3)+a_{2}(3)^{2}+a_3(3)^3=a_{0}+3 a_{1}+9 a_{2}+27a_3=-72}\\ \end{array} $$ The above system gas the solution $$a_{0}=42, \quad a_{1}=-41, \quad a_{2}=-2, \quad a_{3}=1.$$ Hence, $$ p(x)=42-41x-2x^2+ x^3. $$
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