# Find (a) <p,q>, (b) ||p||, (c) ||q||, and (d) d(p,q) for the polynomials in P_2 using the inner product <p,q>=a_0b_0+a_1b_1+a_2b_2.

Find (a) , (b) $||p||$, (c) $||q||$, and (d) $d\left(p,q\right)$ for the polynomials in ${P}_{2}$ using the inner product $.
$p\left(x\right)=1-3x+{x}^{2},q\left(x\right)=-x+2{x}^{2}$

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We have given that.

a) The inner product of the polynomials will be:
$p\left(x\right)=1-3x+{x}^{2},q\left(x\right)=-x+2{x}^{2}$
$
$
$
b) $||p||=\sqrt{{1}^{2}+{\left(-3\right)}^{2}+{1}^{2}}$
$||p||=\sqrt{1+9+1}$
$||p||=\sqrt{11}$
c) $||q||=\sqrt{{\left(-1\right)}^{2}+{2}^{2}}$
$||q||=\sqrt{1+4}$
$||q||=\sqrt{5}$
d) $d\left(p,q\right)=||p-q||$
$d\left(p,q\right)=||1-3x+{x}^{2}-\left(-x+2{x}^{2}\right)||$
$d\left(p,q\right)=||1-3x+{x}^{2}+x-2{x}^{2}||$
$d\left(p,q\right)=||1-2x-{x}^{2}||$
$d\left(p,q\right)=\sqrt{{1}^{2}+{\left(-2\right)}^{2}+{\left(-1\right)}^{2}}$
$d\left(p,q\right)=\sqrt{1+4+1}$
$d\left(p,q\right)=\sqrt{6}$