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How is FOIL used in factoring polynomials?

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FOIL is used to check your answer after you factor a quadratic expression.  For example, if you are asked to factor x2 + x - 12.  The answer would be (x - 3)(x + 4).  To check to see if this is correct, use FOIL; if it is correct, you get back the original expression:

F: (x)(x) = x2 +

O: (x)(4) = 4x +

I: (- 3)(x) = - 3x +

L: (- 3)(4) = - 12

(x - 3)(x + 4) = x2 + x - 12 (original expression)

FOIL can be used to factor quadratic expressions, that is, formulas of the type

ax2 + bx + c

I feel here focus on the simpler type where the first coefficient is 1, that is

x2 + bx + c

This can be factored into the product  (x + d)(x + e)

by realizing that  c = d*e          (the "L" part of FOIL is just a number, no x)
and                   b = d + e        (the "O" and "I" parts of FOIL together give dx + ex, that is (d+e)x)
                                              (the "F" part is this special case is just x*x = x2)

This because of FOIL (try it out by multiplying out (x + d)(x + e).

So, as an example, let's say you need to factor  x2 +  3x  + 2, you know because of FOIL that the product has to be of the shape

(x + d)(x + e)

where  d*e = 2
and      d+e = 3

Can you think of two numbers which multiplied together give 2 and added together give 3?

Yes, 1 and 2! So the result of the factoring is

(x + 1)(x+2)    or     (x + 2)(x+1)

The order makes no difference.

Another example:

Factor  x2  - 5x + 6

Steps: Find two numbers that together multiply to 6 and add to -5.

1. The middle term is negative, that means that at least one of the numbers is negative.
2. The last term is positive, that means that the two numbers have the same sign.
3. Together that means that both numbers are negative.

4. Find two negative numbers which multiply to 6: either -1 and -6 or -2 and -3.
5. Which pair of numbers adds to -5? -2 and -3.

Result: (x - 2)(x - 3)

If you want more examples, or want to know how it works for equations in which x2 has a coefficient other than 1, email me.


FOIL method is used for multiplying binomials. For example

F irst Terms

O utside Terms

I nside Terms

L ast Terms

(3x - 4)(2x + 1)

(3x)(2x) + (3x)(1) + (-4)(2x) + (-4)(1)

6x2 + 3x - 8x - 4

6x2 - 5x - 4 (combine like terms)

Remember FOIL works only when you multiply two binomials

I hope this helps you.