Answer to Question #280859 in Linear Algebra for ASIF

Question

Express V= 2t² + 5t + 9 as a linear combination of the polynomials

P1= t + 1

P2= t - 1

P3 = t² - 2t + 1

Answer 1

Let

$$V= 2t^2 + 5t + 9$$

$$=a(t+1)+b(t-1)+c(t^2- 2t + 1)$$

Then

$$2t^2 + 5t + 9=at+a+bt-b+ct^2-2ct+c$$

$$=ct^2+(a+b-2c)t+(a-b+c)$$

$$t^2:2=c$$

$$t^1:5=a+b-2c$$

$$t^0=9=a-b+c$$

$$c=2$$

$$2a-c=14$$

$$2b-3c=-4$$

$$a=8, b=1, c=2$$

$$V= 2t^2 + 5t + 9$$

$$=8(t+1)+1(t-1)+2(t^2- 2t + 1)$$