Multiplication of Polynomials

Multiplication of polynomials will be done as per the following steps.

Multiply each term in the first polynomial by each term in another polynomial.
Combine the like terms and add them.

Here are examples of multiplication of monomials, binomials as well as the polynomials.

**Monomial with Monomial [ 1 term * 1 term ]**

To multiply one monomial by another monomial, we have to first multiply the constant, then multiply each variable together and then combine the results.

Example: (4xy )(8y)

Solution: (4xy)(8y) = 4.8.xy.y = $12xy^{2}$

Monomial with Binomial [ 1 term * 2 terms]

To multiply one monomial with the binomial term, we have to multiply single term with each of two-term and then combine the results.

Example: 5y( y + 3yz)

Solution: 5y( y + 3yz) = 5y.y + 5y.3yz = $5y^{2} + 15y^{2}z$

Binomial with Monomial [ 2 terms * 1 term]

To multiply binomial term with monomial, just multiply each term of binomial with the single term and then combine the results.

Example: $(2y^{2}-y)3z$

Solution: $(2y^{2}-y)3z$ = $6y^{2}- 3yz$

Binomial with Binomial [ 2 terms * 2 terms]

To multiply two binomial, we have to follow the concept of FOIL (First Outer Inner Last) method and then combine the like terms and add them.

Example: (x+4) (x+8)

The F stands for first, which means the x in the first factor times the x in the second factor.
The O stands for outer, which means the x in the first factor times the 8 in the second factor.
The I stands for inner, which means the 4 in the first factor times the x in the second factor.
The L stands for last, which means the 4 in the first factor times the 8 in the second factor.

Solution: (x+4) (x+8) = $x^{2}+8x+4x+32$ = $x^{2}+12x+32$

Binomial with Trinomial [ 2 terms * 3 terms]

Here FOIL doesn’t work, beacuse there are more terms. Following are the steps to multiply a 2 term polynomial with trinomial term.

Take the first term in polynomial one and multiply it with every term in the second polynomial
Follow the rule of changing sign while multiplying.
Take the second term in polynomial one and multiply it with every term in the second polynomial
Follow the rule of changing sign while multiplying.
Now combine the like terms to obtain the result.

Example: $(x+3) (x^{2}-3x+4)$