Question

The set B={3+1x^2, 9+4x+3x^2, −35−12x−12x^2} is a basis for P_3. Find the coordinates of p(x)= −46−8x−16x^2 relative to this basis

Answer 1

It looks like you need to find constants \(a,b,c\) such that
\[a(3+x^2)+b(9+4x+3x^2)+c(-35-12x-12x^2)=-46-8x-16x^2\]
correct?

Answer 2

If that's the case, then the problem comes down to solving the following system of equations:
\[\begin{cases}
3a+9b-35c=-46&\text{constant terms}\\
4b-12c=-8&\text{linear terms}\\
a+3b-12c=-16&\text{quadratic terms}
\end{cases}\]

Answer 3

@SithsAndGiggles thank you for looking, i tried solving the system and I got a = 236, b = 274, and c = 92, this was not correct :(

Answer 4

How did you work through it? I used a calculator, and indeed that is not the right answer.

Answer 5

@SithsAndGiggles i had -46 instead of -16 for the rhs of the last term, im getting the correct answer now, thank you :)!

Answer 6

No problem! You're welcome