Polynomial Multiplication
To factor polynomials in the second degree, all you need to do is remember
the acronym FOIL. The FOIL method refers to the order that you multiply
the terms. Here is what the acronym stands for:
F—first
O—outer
I—Inner
L—Last
Example: (x + 7) (2x + 5)
Example: (x + 7) (2x + 5)
In the example, the first terms (F) for each set of parenthesis
are x and 2x.
The outer terms (O) are x and 5.
The Inner terms (I) are 7 and 2x.
The last terms (L) are 7 and 5.
To use the FOIL method, multiply each of these pairs of terms.
You will end up with
2x2 + 5x + 14x + 35. Notice that by using the FOIL method,
you automatically end up with a polynomial that is in the proper
descending order. All you have to do to finish the problem is combine
the like terms of 5x + 14x. You will end up with 2x2
+ 19x + 35.
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Example: (3x + 6) (4x + 9)
Here are the terms:
First: 3x and 4x
Outer: 3x and 9
Inner: 6 and 4x
Last: 6 and 9
When you multiply, it will end up as 12x2 + 27x + 24x + 54. This
simplifies to
12x2 + 51x +54.
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Here is an example with negatives:
(4x – 3) (x – 7)
First: 4x and x
Outer: 4x and – 7
Inner: - 3 and x
Last: - 3 and – 7
When you multiply, it will end up as 4x2–28x –3x
+ 21. This simplifies to
4x2– 31x + 21.
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Here is a final example with a positive
and a negative: (5x + 8) (2x – 6)
First: 5x and 2x
Outer: 5x and –6
Inner: 8 and 2x
Last: 8 and –6
When you multiply, it will end up as 10x2 –30x + 16x –48
which simplifies to
10x2 – 14x –48.
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Patterns
Difference of Squares
Some polynomials have patterns that make finding the answer a little
bit easier. One of the easiest becomes a difference of squares. Here is
an example.
Example: (x – 3) (x
+ 3)
If you start with (x – 3) (x + 3) and use the FOIL method,
you end up with
x2 + 3x – 3x – 9. When you simplify this
you end up with x2– 9, or the difference of two
perfect squares. Any time the terms in your factors are the same,
only the signs are opposite, they will multiply out to be a difference
of the two terms squared. You can save a step if you know this trick.
Here are some more examples:
(2x – 4) (2x + 4)
This problem follows the pattern. The terms are the same, but the
signs are different. Instead of using the FOIL method, you can find
this using the shortcut. All you have to do is square the two terms
and then subtract. You end up with:
4x2– 16
Notice that you squared 2x and 4, and subtracted the squares. This
always works for problems that follow this pattern. Here is one
more example:
(5x + 9) (5x – 9) = 25x2– 81
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Perfect Square Trinomials
If you square two terms in a set of parenthesis, you will end up with
what is called a perfect square of trinomials. There is a shortcut as
well for finding a perfect square trinomial. Here is an example:
(3x + 4)2
With the FOIL method, you get 9x2 + 12x + 12x +16, which simplifies to
9x2 + 24x + 16.
Notice that 9x2 and 16 are the perfect squares of the terms in the problem
(3x and 4). So you can find the first and last terms by taking the perfect
squares of the terms in the problem. To find the middle term, multiply
the two terms together and double the answer. 3x times 4 is 12x, and doubled
is 24x, which is our middle term.
Example: (x – 6)2
Square both terms: x2 - ______ + 36.
Multiply the terms together and double: x times –6 is –6x,
and doubled is -12x. The answer is x2 – 12x + 36. Notice that
because 6 is negative, the middle term is negative.
(4x + 15)2
Square both terms: 16x2 + __________ + 225
Multiply the terms together and double: 120x. Answer: 16x2
+ 120x + 225.
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Example: (3x – 10)2
Square both terms: 9x2 - __________ + 100
Multiply the terms together and double: -60x. Answer: 9x2–
60x + 100. |
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