# General Formula for Multiplying Polynomials?

• kpoltorak
In summary, the equation relates the coefficients of a product of two sequences, where each sequence is the discrete convolution of the other sequence and the product is itself the discrete convolution of the two sequences.
kpoltorak

## Homework Statement

Does a general formula exist?
$$\sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k$$

## The Attempt at a Solution

I am having trouble understanding the relation between the c coefficients in the product and the respective a and b coefficients in the factors. Could somebody shed some light on this? I am fairly certain there is a relation between the two which is based on combinatorics but cannot find the answer anywhere.

See if you can solve an "easier" problem. Try assuming m1=m2 (if you get this the general case isn't hard to handle).

Look at a specific case m1 = m2 = 2.

(a0 +a1x +a2x^2) * (b0 +b1x + b2x^2) =

(a0b0)x^0 + (a1b0 + b1a0)x + (a2b0 +a1b1 + a0b2)x^2 + (a1b2 + a2b1)x^3 + (a2b2)x^4

Do you see how to write this as a summation?

I know am digging this thread out of the bottom of the universe but Google got me here, and am sure more people could use this answer.

I am writing a program for multiplying polynomials of equal length and need a general formula for the coefficients in the resulting polynomial. I literary stumped as I don't see much of a pattern here. Any one have a clue?

Look up the Cauchy Product. One place is here:
http://www.astarmathsandphysics.com/university_maths_notes/analysis/university_maths_notes_analysis_cauchy_products_of_series.html

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Thank you :)

kpoltorak said:

## Homework Statement

Does a general formula exist?
$$\sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k$$

## The Attempt at a Solution

I am having trouble understanding the relation between the c coefficients in the product and the respective a and b coefficients in the factors. Could somebody shed some light on this? I am fairly certain there is a relation between the two which is based on combinatorics but cannot find the answer anywhere.

The sequence $\{ c_k \}$ is the (discrete) convolution of the two sequences $\{ a_k \} \text{ and } \{ b_k \}.$ See, eg., http://www.cg.tu-berlin.de/fileadmin/fg144/Courses/07WS/compPhoto/Convolution_charts.pdf .

RGV

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## 1. What is the general formula for multiplying polynomials?

The general formula for multiplying two polynomials is (anxn + an-1xn-1 + ... + a1x + a0) * (bmxm + bm-1xm-1 + ... + b1x + b0) = cn+mxn+m + cn+m-1xn+m-1 + ... + c1x + c0, where ci = ai*bi for all i.

## 2. What are polynomials?

Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but not division. They can have multiple terms, each containing a variable raised to a different power.

## 3. How do you multiply polynomials with more than two terms?

To multiply polynomials with more than two terms, you can use the distributive property. First, multiply each term in the first polynomial by each term in the second polynomial. Then, combine like terms by adding the coefficients of terms with the same variable raised to the same power.

## 4. Can you explain the FOIL method for multiplying polynomials?

The FOIL method is a mnemonic device used to remember the steps for multiplying two binomials. It stands for First, Outer, Inner, Last. First, multiply the first terms of each binomial. Then, multiply the outer terms, followed by the inner terms, and finally the last terms. Finally, add all the resulting terms together to get the final product.

## 5. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, in the polynomial 3x2 + 5x + 1, the degree is 2 because that is the highest power of x. The degree is important in determining the behavior and properties of a polynomial.

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