Question

Express the following as a linear combination of

p1 = 3 + x + 2x2, p2 = 5 - x + 5x2, and p3 = 2 + 5x + 3x2.

Answer 1

$p_1=2x^2+x+3$

$p_2=5x^2-x+5$

$p_3=3x^2+5x+2$

$x^2+x+1=a_1(2x^2+x+3)+b_1(5x^2-x+5)+c_1(2+5x+3x^2)$

$x^2(2a_1+5b_1+3c_1)+x(a_1-b_1+5c_1)+(3a_1+5b_1+2c_1)=x^2+x+1$

$2a_1+5b_1+3c_1=1;a_1-b_1+5c_1=1;3a_1+5b_1+2c_1=1$

$(3a_1+5b_1+2c_1)-(3a_1+4b_1+8c_1)=-1$

$b_1=6c_1-1$ ---(1)

$2(b_1-5c_1+1)+5b_1+3c_1=1$

$7b_1-6c_1+1=0$ ---(2)

Solving (1) and (2) we get

$c_1=6/35$

Substituting this in equation(1) we get

$b_1=1/35;a_1=6/35$

So,

$x^2+x+1=(6/35)(2x^2+x+3)+(1/35)(5x^2-x+5)+(6/35)(2+5x+3x^2)$