MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL
As with the monomial multiplier, we explain multiplication of a polynomial by a
polynomial by use of an arithmetic example. To multiply (3 + 2)(6  4), we could do the
operation within the parentheses first and then multiply, as follows:
(3 + 2)(6  4) = (5)(2) = 10
However, thinking of the quantity (3 + 2) as one term, we can use the method described for
a (3 + 2), with the following result:
(3 + 2)(6  4) = [(3 + 2) x 6  (3 + 2) x 4]
Now considering each of the two resulting
products separately, we note that each is a
binomial multiplied by a monomial.
The first is
(3 + 2)6 = (3 x 6) + (2 x 6)
and the second is
(3 + 2)4 =  [(3 x 4) + (2 x 4)]
= (3 x 4)  (2 x 4)
Thus we have the following result:
(3 + 2)(6  4) = (3 X 6) + (2 X
6)
(3 x 4)  (2 x 4)
= 18 + 12  12  8
= 10
The complete product is formed by multiplying each term of the multiplicand separately by
each term of the multiplier and combining the
results with due regard to signs.
Now let us apply this method in two examples involving literal numbers.
1. (a + b)(m + n) = am + an + bm + bn
2. (2b + c)(r + s + 3t  u) = 2br + 2bs + 6bt  2bu + cr + cs + 3ct  cu
The rule governing these examples is stated as follows: The product of any two polynomials is
found by multiplying each term of one by each
term of the other and adding the results algebraically.
It is often convenient, especially when either of the expressions contains more than two
terms, to place the polynomial with the fewer
terms beneath the other polynomial and multiply term
by term
beginning at the left. Like 3x^{2}  7x  9 and 2x  3. The procedure is
Practice problems. In the following problems, multiply and combine like terms:
SPECIAL PRODUCTS
The products of certain binomials occur frequently. It is convenient to remember the form
of these products so that they can be written
immediately without performing the complete multiplication process. We present four such
special products as follows, and then show how
each is derived:
1. Product of the sum and difference of two numbers.
EXAMPLE: (x  y)(x + y) = x^{2}  y^{2}
2. Square the sum of two numbers.
EXAMPLE: (x+y)^{2}=x^{2} +2xy+y^{2}
3. Square of the difference of two numbers.
EXAMPLE: (x  Y)^{2} = x^{2} 2xy+y^{2}
4. Product of two binomials having a common term.
EXAMPLE: (x + a)(x + b) = x^{2} + (a + b)x + ab
Product of Sum and Difference
The product of the sum and difference of two numbers is equal to the square of the first
number minus the square of the second number.
If, for example, x  y is multiplied by x + y, the
middle terms cancel one another. The result is the square of x minus the square of y, as
shown in the following illustration:
By keeping this rule in mind, the product of the
sum and
difference of two numbers can be written down immediately by writing the
difference of the squares of the numbers. For example, consider the following three problems:
