document.write( "Question 1139652: Show that P³= span {x²+x³, x, 2x²+1,3} \n" ); document.write( "

Algebra.Com's Answer #760169 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "I am only vaguely familiar with this subject; so my response might not be mathematically rigorous.

\n" ); document.write( "We are to show that every polynomial of degree 3 (element of P³) can be formed by a linear combination of $\"x%5E3%2Bx%5E2\"$ , $\"2x%5E2%2B1\"$ , $\"x\"$ , and $\"3\"$ .

\n" ); document.write( "Let an arbitrary third degree polynomial be $\"ax%5E3%2Bbx%5E2%2Bcx%2Bd\"$ . Show that the polynomial can be expressed as $\"p%28x%5E3%2Bx%5E2%29%2Bq%282x%5E2%2B1%29%2Br%28x%29%2Bs%283%29\"$ .

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\n" ); document.write( "We need to show that we can express the arbitrary coefficients a, b, c, and d in terms of p, q, r, and s. It is easy to see that this is possible by equating the coefficients of each term in the two polynomials:

\n" ); document.write( " $\"ax%5E3%2Bbx%5E2%2Bcx%2Bd+=+%28p%29x%5E3%2B%28p%2B2q%29x%5E2%2B%28r%29x%2B%28q%2B3s%29\"$

\n" ); document.write( "a = p; b = p+2q; c = r; d = q+3s

\n" ); document.write( "Example: $\"7x%5E3-5x%5E2%2B4x-3\"$ (a = 7; b = -5; c = 4; d = -3)

\n" ); document.write( "a = p = 7;
\n" ); document.write( "b = p+2q = 7+2q = -5 --> q = -6;
\n" ); document.write( "c = r = 4;
\n" ); document.write( "d = q+3s = -6+3s = -3 --> s = 1.

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