Check Out This Example of Polynomial With Solution

Task:

Check whether the polynomials p1(x)=x2+4x, p2(x)=1+x3+x2, p3(x)=x3+x, p4(x)=x2-5 form a basis in the space P3.

Solution:

The standard basis in the space P3 look like {1, x, x2, x3}, and they have 4 basis elements. To prove that polynomials p1, p2, p3, p4 form a basis in the space P3 it is sufficient to show that they are linearly independent.

Let’s form a linear combination:

C1 p1 ( x) + C2 p2 ( x) + C3 p3 ( x) + C4 p4 ( x) = C1  (x 2 +4 x) +  C2  (1 +  x 3 +x 2 ) + C3  (x 3 + x) + C4 ( x 2 — 5) = C1 x 2 + C1 4 x  + C2  + C2 x 3 + C2 x 2  + C3 x 3 + C3 x + C4 x 2  – C4 5 =

x 3 (C2 + C3 ) + x 2  (C1 + C2 + C4 ) +  x (4C1 +  C3   +  C2 – 5C4 )

This linear combination is equal to zero only if all coefficients are zero at powers of x, so we have come to the system:

The system’s only solution is C1 = C2=C3 = C4 = 0 which means that the polynomials  p1, p2, p3, p4 are linearly independent and therefore form a basis in the space P3.

 

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