When multiplying a polynomial by a monomial, the distributive property and rules of
exponents are used.
example 1
Multiply: 4x2 · (2x + 3x3)
When two polynomials are multiplied together, each term from the first polynomial must
be multiplied by each term of the second polynomial.
example 2
Multiply: (x + 2y)(3x – 4y)
Another method for multiplying the polynomials in this example is called the FOIL
method.
When multiplying two binomials (polynomials with two terms), the FOIL method works
as follows:
example 3
Multiply: (x + 2y)(3x – 4y)
The FOIL method is useful for making sure no steps are overlooked, but it is only used
when multiplying binomials.
When multiplying any other polynomials, remember to multiply every term of the first
polynomial by every term of the second polynomial.
example 4
Multiply the polynomials:
Special Products
There are some special cases involving the multiplication of two binomials. The FOIL
method can be used anytime two binomials are multiplied, but understanding the
following rules will make it easier to understand future lessons about factoring
polynomials.
Square of a Sum
When a binomial is squared, the resulting polynomial is the first term of the binomial
squared, twice the two terms multiplied together, and the last term of the binomial
squared.
This can be proven with the FOIL method:
This shortcut can be used to skip a step or two in the FOIL method.
example 5
Multiply:
Square of a Difference
This is the same as the square of a sum, except that a minus sign appears ahead of the
middle term of the resulting trinomial.
example 6
Multiply:
Difference of Squares
The difference of squares refers to the answer that results from multiplying two
binomials with the same terms, but with the last term of each binomial having opposite
signs:
The answer has no middle term because the like terms that result from the FOIL method
cancel each other out:
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Name ______________________________
Find each product.
1) 2x(3x – 7)
2) −5x3(6x2 + 2y2)
3) a2b4(a2b – 3ab2)
4)
5) 5b(2b3 – 4b2 + 3b)
6) 2x3y(4x3y – x2y2 + 3xy3)
7) (x + 6)(x – 3)
8) (2y – 3)(y + 7)
9) (a – 2b)(4a + b)
10) (3m2 + n)(m – 4n)
1 3
mn
3
Find each product.
11) (x + 3)(x2 – 3x + 2)
12) (3c – 2)(6c2 + 5c – 11)
13) (m + n)(m + n)
14) (2p + 3q)(2p + 3q)
15) (3c – d)(3c – d)
16) (5g – 2h)(5g – 2h)
17) (x + y)(x – y)
18) (3a + 4b)(3a – 4b)
19) (2m2 – 1)(2m2 + 1)
20)