Fp(x) and q(x) are arbitrary polynomials of degreeat most 2, then the mapping< p, q > = p(-2)q(-2)+ p(0)q(0)+ p(2)q(2)defines an inner product in p3.use this inner product to find < p, q > , llpll, llqll, and the angletetha, between p(x) and q(x) forp(x) = 2x^2+6x+1 and q(x) = 3x^2-5x-6.< p; q > = ? llpll = ? llqll= ?

We're given an inner product defined by

That is, we multiply the values of and at and add those products together.

The inner product is

To find the norms and , recall that the dot product of a vector with itself is equal to the square of that vector's norm:

So we have

Finally, the angle between and can be found using the relation

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."

5x+10 is the correct one

hope this