Consider the following polynomials in P3: q(x)=−x3+3x2−x+5 r(x)=−4x3+7x2−x+10 u(x)=−5x3+8x2+10 For each of the following polynomials, determine whether it is in {q,r,u}. If so, ecpress it as a lineare combination of the polynomials above. Use e.g. q rather than q(x) to represebt the polynomials in your linear combination. p1(x)=−3x2−3x+10

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okomgcae · Accepted Answer

Given the polynomials q(x)=−x3+3x2−x+5,r(x)=−4x3+7x2−x+10 and u(x)=−5x3+8x2. Now, a polynomial p(x) is in span of {q,r,u} if there exist nonzero constants a,b and c such that p(x)=aq+br+cu The polynomial is p1(x)=−3x2−3x+10 Then, p1=aq+br+cu −3x2−3x+10=a(−x3+3x2−x+5)+b(−4x3+7x2−x+10)+c(−5x3+8x2+10)−3x2−3x+10 =(a−4b−5c)x3+(3a+7b+8c)x2+(−a−b)x+5a+10b+10c Equate the coefficients of both sides and find the values of a, b and c. a−4b−5c=0 (1) 3a+7b+8c=−3 (2) −a−b=−3 (3) 5a+10b+10c=10 (4) From equation (3), a=−3+b a=3−b Substitute b=3−a in equation (1) and equation (2) and (3) −(3−b)−4b−5c=0 (4) 3(3−b)+7b+8c=−3 (5) 5(3−b)+10b+10c=10 (6) From equation (4), −3+b−4b−5c=0 −3−3b−5c=0 −3b=5c+3 b=−5c+33 Substitute b=−5c+33 and in equation (5) and (6) −9+4(−5c+33)+8c=−3 (7) 15+5(−5c+33)+10c=10

Consider the following polynomials in P_3:q(x)=-x^3+3x^2-x+5, r(x)=-4x^3+7x^2-x+10, u(x)=-5x^3+8x^2+10

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