Question

Suppose that f(x) and g(x) are given by the power series:

f(x)= 4+ 2x + 4x^2 + 2x^3+⋯
and
g(x)= 2+ 5x + 3x^2 + 3x^3+⋯
By multiplying power series, find the first few terms of the series for the product
h(x) = f(x) * g(x)= c0 + c1x + c2x^2 + c3x^3+⋯

Answer 1

• c0 = 8
• c1 = 24
• c2 = 30
• c3 = 42

Explanation:

It is convenient to notice that the pattern of coefficients that need to be multiplied and summed is a pattern of Xs.

For the constant, c0, only the constants need to be multiplied.

c0 = 4·2 = 8

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For the x-term coefficient, the constant of one series needs to be multiplied by the x-term of the other, and the two products added.

c1 = 4·5 +2·2

c1 = 24

If you draw real or imagined lines between the coefficients being multiplied, you see that they form an X.

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For the x²-term coefficient, the above X gets wider, and a vertical | is added. That is constant multiplies x² term and x-term multiplies x-term. The sum has three terms:

c2 = 4·3 +2·4 +2·5

c2 = 30

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For the x³ term, there are now two Xs, one wide and one narrower. Constant terms multiply x³ terms and x-terms multiply x² terms.

c3 = 4·3 +2·2 +2·3 +5·4

c3 = 42

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This sort of pattern of Xs can be used wherever polynomials (or multidigit numbers) are multiplied. It can help you ensure that all of the products that need to be included in a sum are accounted for.