Repeat Exercise 1 using the polynomial algebra method indicated in Exercise 2.Exercise 1Fi...

Repeat Exercise 1 using the polynomial algebra method indicated in Exercise 2.

Exercise 1

Find the coordinate vector of the polynomial 4x3 − 9x2 + x relative to the ordered basis B′ = ((x − l)3, (x − l)2, (x − 1), 1) of the vector space P3 of polynomials of degree at most 3. Use the method illustrated in Example 1.

Exercise 2

Example 1 showed how to use linear algebra to rewrite the polynomial p(x) = x3 + x2 − x − 1 in powers of x + 1 rather than in powers of x. This exercise indicates a polynomial algebra solution to this problem. Replace x in p(x) by [(x + 1) − 1], and expand using the binomial theorem, keeping the (x + 1) intact. Check your answer with that in Example 1.

EXAMPLE 1

Find the coordinate vector of p(x) = x3 + x2 − x − 1 relative to the ordered basis B′ = ((x + 1)3, (x + 1)2, x + 1, 1).

SOLUTION

Multiplying out the powers of x + 1, to express the vectors in B′ in terms of our usual ordered basis B = (x3, x2, x, 1) for P3, we see that

 B′ = (x3 + 3x2 + 3x + 1, x2 + 2x + 1, x + 1, 1).

Using coordinates relative to the ordered basis B, our problem reduces to expressing the vector [1, 1, −1, −1] as a linear combination of the vectors [1, 3, 3, 1], [0, 1, 2, 1], [0, 0, 1, 1], and [0, 0, 0, 1]. Reducing the matrix corresponding to the associated linear system, we obtain

Thus the required coordinate vector is p(x)B′ = [1, −2, 0, 0], and so

 x3 + x2 − x − 1 = (x + l)3 − 2(x + l)2.

Linear algebra is not the only tool that can be used to solve the problem in Example.