When multiplying two polynomials together, you are basically performing the distributive property from one polynomial through to the other.
For example, if I wanted to multiply these two binomials:
(2x + 3) (4x + 5)
I would distribute (multiply through a set of parentheses) the 2x to the 4x +5 in the other parentheses, and then perform the same distributive property with the 3 to the same 4x + 5. I would then be left with:
2x(4x + 5) + 3(4X + 5)
8X2 + 10X + 12X + 15
Typically, the middle two terms can be added together, as they will often be like terms. I will arrive at:
8X2 + 22x + 15
When specifically multiplying two binomials (which is the most common multiplying task you will find in your coursework), this multiplication process is often referred to “FOIL” ing: First, Outer, Inner, Last. This identifies the terms you can multiply in order to distribute the terms properly.
First: Multiply the first term of each binomial together. 2x∙4x
Outer: Multiply the outer terms together (the first term of the first binomial times the last term of the second binomial) 2x∙5
Inner: Multiply the inner terms together (the second term of the first binomial times the first term of the second binomial) 3∙4x
Last: Multiply the last term of each binomial together. 3∙5
You can see we will arrive at the same four terms we got when we distributed above (before we combined the middle two terms), but we seem to like the acronym FOIL as a name instead of “double distributive property.”
Example using the FOIL
Here’s a link to a video showing you how to do another example of the FOIL method.
Practice Problems: Now you try!
- (x + 3)(x + 6)
- (2x + 5)(x + 3)
- (7x – 2)(4x + 3)
- (9x + 1)(3x – 8)
- (5x2– 6)(x2– 6)
- (4w – y)(4w + y)
These middle terms canceled each other out.
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