Let $P_{2}$ $P_2$ is real polynomial space with the highest degree is 2. Specify the $x^{2} -coordinate$ $x^2"-coordinate "$ of the base ${x^{2} + x, x + 1, x^{2} + 1} in P_{2}$ $" "{x^2+x,x+1,x^2+1}" in "P_2$ ?

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Answer 1 · 2017-04-09T09:10:52

$(\frac{1}{2}, - \frac{1}{2}, \frac{1}{2})$ $(1/2,-1/2,1/2)$

I think you are asking for the coordinates of $x^{2}$ $x^2$ using the given base.

We find:

$\frac{1}{2} (x^{2} + x) - \frac{1}{2} (x + 1) + \frac{1}{2} (x^{2} + 1) = x^{2}$ $1/2(x^2+x)-1/2(x+1)+1/2(x^2+1) = x^2$

So the coordinates are: $(\frac{1}{2}, - \frac{1}{2}, \frac{1}{2})$ $(1/2, -1/2, 1/2)$

Answer 2 · 2017-04-09T09:13:27

$x^{2} = \frac{1}{2} (x^{2} + x) - \frac{1}{2} (x + 1) + \frac{1}{2} (x^{2} + 1)$ $x^2 = 1/2(x^2+x)-1/2(x+1)+1/2(x^2+1)$

We have that

$x^{2} = α (x^{2} + x) + β (x + 1) + γ (x^{2} + 1)$ $x^2=alpha(x^2+x)+beta(x+1)+gamma(x^2+1)$

so

$x^{2} = (α + γ) x^{2} + (α + β) x + (β + γ)$ $x^2=(alpha+gamma)x^2+(alpha+beta)x+(beta+gamma)$

and we need

${(alpha+gamma=1),(alpha+beta=0),(beta+gamma=0):}$

solving we have

$alpha=1/2,beta=-1/2,gamma=1/2$ and finally

$x^2 = 1/2(x^2+x)-1/2(x+1)+1/2(x^2+1)$

Let P_2P2P_2 is real polynomial space with the highest degree is 2. Specify the x^2"-coordinate "x2-coordinate x^2"-coordinate " of the base " "{x^2+x,x+1,x^2+1}" in "P_2 {x2+x,x+1,x2+1} in P2" "{x^2+x,x+1,x^2+1}" in "P_2 ?