Express V= 3t² + 7t + -4 as a linear combination of the polynomials
P1= t² + 2t + 3
P2= 2t² + 3t + 7
P3 = 3t² + 5t + 6
1
Expert's answer
2021-08-02T14:46:08-0400
Let us express V=3t2+7t−4 as a linear combination of the polynomials
P1=t2+2t+3,P2=2t2+3t+7,P3=3t2+5t+6.
Let V=aP1+bP2+cP3.
Then
3t2+7t−4=a(t2+2t+3)+b(2t2+3t+7)+c(3t2+5t+6).
It follows that
3t2+7t−4=at2+2at+3a+2bt2+3bt+7b+3ct2+5ct+6c,
and hence
3t2+7t−4=(a+2b+3c)t2+(2a+3b+5c)t+(3a+7b+6c).
Therefore, we have the following system:
⎩⎨⎧a+2b+3c=32a+3b+5c=73a+7b+6c=−4
Let us multiply the first equation by −2 and add to the second equation, also multiply the first equation by −3 and add to the third equation. Thgen we have the system:
⎩⎨⎧a+2b+3c=3−b−c=1b−3c=−13
Let us add the last to equation. Then we get the system:
Comments