Polynomial multiplication is a very common operation throughout Algebra and Mathematics in general. We use following three properties very frequently all the way along as we work on multiplication of polynomials.
- Associative Properties of Algebra
- Distributive Properties of Algebrat
- Commutative Properties of Algebra
- Rules / Laws of Exponents
We have already discussed the first three properties in the preceding topics.
We enlist the relevant formulas below so that you can revise these properties quickly:
Commutative Law For Addition
`a + b = b + a`
Commutative Law For Multiplication
`a*b = b * a`
Associative Law For Addition
`a + ( b + c ) = ( a + b ) + c`
Associative Law For Multiplication
`a * ( b *c ) = ( a * b ) * c`
`a *( b + c ) = ( a* b ) + ( a * c )`
What remains is the rules of exponents. We explain these rules first and then we move forward to polynomial multiplication.
Let a be a real number. And let m,n be any positive integers. Then:
`a^m * a^n = a^m+n `
If a is any real number, and m,n are two positive integers:
` (a^m)n = a^mn `
Let a,b be two real numbers. And let n be a positive integer then:
` (ab)^n = a^n.b^n`
These properties are very simple and easy to verify. You must learn them by heart before you move ahead towards the multiplication of polynomials.
These are some of the most basic rules for polynomial multiplication. Now that you have practiced these simplifications, the process of multiplication is going to be very easy for you
The simple rule that we use while multiplying two polynomials is that
While multiplying these terms, we make use of the rules of exponents stated above, whereas the remaining formulas help us simplify and / or expand the multiplying polynomials.
We start with the simpler examples (involving monomials) and then proceed towards more complex examples (those involving polynomials with 2 and more terms).
FOIL Method is a reserved word for a method of multiplying two binomials. The four letters of the word FOIL each refer to the following four operations:
We solve a few examples of binomial multiplication using FOIL method: