What is "multiplying polynomials"?

Expert Answers
cburr eNotes educator| Certified Educator

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.

atyourservice | Student

Multiplying polynomials means to multiplying numbers with variables and coefficients

For example:

(x + 3) (x + 5)

You foil and multiply x by x and 5 and then multiply 3 by x and 5

x^2  + 5x + 3x + 15


x^2 + 8x +15

is an example on how you should multiply polynomials:

princessenotes | Student

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

giorgiana1976 | Student

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.


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





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:


2. The multiplication is associative.


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


4. The mutiplication is distributive with respect to addition.


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