Find (a) , (b) ||p||, (c) ||q||, and (d) d(p,q) for the polynomials in P2 using the inner product <p,q≥a0b0+a1b1+a2b2. p(x)=1−3x+x2,q(x)=−x+2x2
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We have given that. p(x)=1−3x+x2, q(x)=−x+2x2 a) The inner product of the polynomials will be: p(x)=1−3x+x2,q(x)=−x+2x2 <p,q≥1×0+(−3)(−1)+1(2) <p,q≥3+2 <p,q≥5 b) ||p||=12+(−3)2+12 ||p||=1+9+1 ||p||=11 c) ||q||=(−1)2+22 ||q||=1+4 ||q||=5 d) d(p,q)=||p−q|| d(p,q)=||1−3x+x2−(−x+2x2)|| d(p,q)=||1−3x+x2+x−2x2|| d(p,q)=||1−2x−x2|| d(p,q)=12+(−2)2+(−1)2 d(p,q)=1+4+1 d(p,q)=6
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(2x−1) is a factor of the polynomial 6x6+x5−92x4+45x3+184x2+4x−48. Determine whether the statement is true or false. Justify your answer.
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