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Foil Method

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In algebra, to multiply two binomials, a method is defined called as FOIL method. FOIL stands for "first, outer, inner and last term".
Here,
First means initial terms of each binomial are multiplied together.
Outer means initial term of binomial is multiplied by second term of other binomial.
Inner means inside terms or numbers are multiplied, which means first term of initial binomial and second term of other binomial are multiplied together.
Last means final term of both binomials are multiplied.

FOIL is an acronym to remember a set of rules for multiplying binomials and is considered as a special case of a more general method for multiplying algebraic expressions using the distributive law. It is used for a two term polynomial times another two term polynomial.

Foil Method Definition

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Foil method definition states that, it is a standard method to multiply two binomials. Binomials are expressions which contain two terms.

When we multiply two binomials, we will get four terms:
  • First: Multiply the first terms of both binomials.
  • Outside: Multiply first term of first binomial with last term of second binomial.
  • Inner: Multiply last term of first binomial and first term of second binomial.
  • Last: Multiply last terms of both the binomials.
Simplify the products and combine the like terms which may occur.

Reverse Foil Method

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Reverse foil method is a process of doing foil backward. It is used to write the information in a more organized way and helps us to understand how FOIL works, when multiplying two binomials.

Reverse foil method of factoring trinomials is in the form x$^{2}$ + bx + c. Through this method, we can convert the trinomial back into a double binomial expression.

Steps to be considered are given below :
  1. Set two sets of parentheses for trinomial equation like (_ $\pm$ _) (_ $\pm$ _).
  2. Factor the given polynomial.
  3. Find the possible factors of the last term.
  4. Plug in the obtained number groups in the given equation. Check whether the obtained factors hold true or not in the given equation.
  5. Keep plugging the factors and check it with the FOIL. If the factors hold true in the given equation, then you've got the answer.

Foil Method Polynomials

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Polynomial is an expression constructed from variables and constants. Expressions with two or more terms are called polynomials. A polynomial can have constants, variables and exponents. But, division and square root is not allowed. They are written in decreasing order of terms. Highest exponent in the polynomial is usually written first.

Polynomial equation also known as algebraic equation is of the form $a_{n}x^{n}$ + $a_{n-1}x^{n-1}$ + .......+ $a_{1}$x + $a_{0}$ = 0

Solved Example

Question: Solve (2x + 5) (x$^{3}$ + 10x$^{2}$ + 8x + 3).
Solution:
Given (2x + 5) [(x$^{3}$ + 10x$^{2}$) + (8x + 3)]
The second expression can be split as (x$^{3}$ + 10x$^{2}$) + (8x + 3), so that, we can apply FOIL.
It can be solved now as follows:
(2x + 5)( (x$^{3}$ + 10x$^{2}$) + (2x + 5) (8x + 3)                    (Use FOIL again)
= 2x(x$^{3}$) + 2x(10x$^{2}$) + 5x$^{3}$ +  50x$^{2}$ + 2x(8x) + 2x(3) + 5(8x) + 5(3)
= 2x$^{4}$ + 25x$^{3}$ + 66x$^{2}$ + 46x + 15

An expression with three terms is called a trinomial. Foil method cannot be applied to trinomials, as it holds only for binomials. Given below are some examples which can be solved in this way. 3p + q - 5xty, 2s - 9t + 8t, 16y - 5z + 1 are all trinomials.

Solved Example

Question: Solve (5x - 2) (5x$^{2}$ + 7x - 5)
Solution:
5x (5x$^{2}$) + (5x) (7x) + 5x (-5) - 2 (5x$^{2}$) - 2 (7x) - 2 (-5)
= 25x$^{3}$ + 35x$^{2}$ - 25x - 10x$^{2}$ - 14x + 10
= 25x$^{3}$ + 25x$^{2}$ - 39x + 10

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Foil Method Examples

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Given below are some of the examples on FOIL method.

Solved Examples

Question 1: Multiply (5x - 4) (x - 4) using FOIL method.
Solution:
First: $5x \times x = 5x^{2}$
Outside: $5x \times (-4) = -20x$
Inside: $- 4 \times x = -4x$
Last: $(-4) \times (-4) = 16$

Summing it all up, we get the trinomial equation as $5x^{2} - 24x + 16$

Question 2: Solve x$^{2}$ + 13x + 36 using reverse FOIL method.
Solution:
Step 1:
Set two sets of parentheses for trinomial equation like: (_ $\pm$ _) (_ $\pm$ _).
Coefficient of first term is 1. So, the binomials of the first terms are x and x.
So, we have, (x $\pm$ _) (x $\pm$ _).

Step 2:
Factor the given polynomial.
As the last sign is positive, find the factors of 36 that add to give 13.

Step 3:
Possible factors of the last term '36' are (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

Step 4:
As 4 and 9 only add to give 13. These factors are the last numbers in the binomial factors as the other factors doesn't hold true for the given equation (x + 4) (x + 9).