What is "multiplying polynomials"?

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Let's have two polynomials, f and g, written in this way:

f=(a0,a1,a2,.....,ak,…an,...), g=(b0,b1,b2,…,bm,....)

where a0, a1, a2....an, b0, b1,....bn are coefficients of these two polynomials.

We can write the polynomials in this way also:

f(x)=a0X^n + a1X^(n-1) + .......+anX^0

g(x)=b0X^m + b1X^(m-1) + .......+bmX^0

These two polinoms have a finite number of terms, different from 0 value. We can define on the set of complex numbers the following algebraical operations: addition and multiplication.

f+g=(a0+b0,a1+b1,a2+b2,...)

fg=(c0,c1,c2,...), (2)

where

c0=a0*b0,

c1=a0*b1+a1*b0,

c2=a0*b2+a1*b1+a2*b0,

ar=a0*br+a1*br-1+a2*br-2+...+ar*b0= ai*br-i= ai*bj

The element f+g=(a0+b0,a1+b1,....) is called the sum between f and g and the operation is called addition.

The element f*g=(c0,c1,c2,....) is called the product between f and g,and the operation is called multiplication.

For example:

If f=(-1,2,3,-5,0,0,..) and g=(1,0,-1,0,...), then their sum is f+g=(0,2,2,-5,0,0,...), and their product is f*g=(-1,2,4,-7,-3,5,0....).

The properties of polynomial multiplication:

1. Commutation between the factors of mutiplying does not change the result:

f*g=g*f

2. The multiplication is associative.

(f*g)*h=f*(g*h)

3. The polynomial 1=(1,0,0,...) is neutral element for multiplication.

f*1=1*f=f

4. The mutiplication is distributive with respect to addition.

f*(g+h)=f*g+g*h

(f+g)*h=f*h+g*h5. f*g=f*h and fis non-zero polynomial ,then we can simplify with f and the result is g=h

**Here are some examples:**

(x + y) (a + b)

x(a + b) = ax + bx

y(a + b) = ay + by

ax + bx + ay + by

The expression is simplified.

**I hope I helped you. If you need anymore help, you can email my teacher at helpmathtutor@gmail.com**

The trick to multiplying polynomials is that every element of one polynomial has to be multiplied against every element of the other.

**1) The clearest way to show this is:**

(x + y) (a + b)

First multiply the second polynomial by the x:

x(a + b) = ax + bx

Then multiply by the y:

y(a + b) = ay + by

Put them all together for the final answer: ax + bx + ay + by

There is no way to simplify this any further.

**2) What if you have the same letters in both?**

(x + y) (x + y) =

x(x + y) + y(x + y) = x^2 + xy + xy + y^2

This can be simplified because you have two xy's

x^2 + 2xy + y^2

**3) Be careful to check whether the numbers are positive or negative:**

In the problem (x + y) (x - y) , all the letters are positive except the second y, so:

(x + y) (x - y) = x (x - y) + y (x - y) = x^2 - xy + xy - y^2

This simplifies to x^2 - y^2 since the -xy and +xy cancel each other out.

These same principles apply no matter how many elements you have in each polynomial.

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