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Calculating the reverse FOIL factorization of a trinomial may be the quickest way to factorize a second-degree trinomial in a single indeterminate. Keep reading this comprehensive article to learn:

  • What is the reverse FOIL method?
  • How do we calculate the reverse FOIL on a generic polynomial?
  • An example of the reverse FOIL in action.

and much more!

Factoring polynomials

A polynomial is an expression where we can identify a set of indeterminates and coefficients, where we can use the following operations:

  • Addition and subtraction (algebraic sum);
  • Multiplication of coefficients and indeterminates; and
  • Exponentiation with non-negative, integer exponents.

Big words, we know, but all will make sense soon.

Polynomials may have more than one type of indeterminates. However, we often first meet polynomials with a single variable in the form of simple equations. Let's see our first polynomial and analyze its elements!

4x2\footnotesize 4x-2

What can we learn from this simple expression:

  • There is a single indeterminate, xx;
  • The indeterminate has exponents 11 (for 4x=4x14x = 4x^1) and 00 (for the constant 2=2x0-2 = -2x^0);
  • The coefficients are the integer numbers 44 and 2-2.
  • As it has only two terms, we call this polynomial a binomial.

This is an example of first-degree polynomial since the indeterminate xx is only ever raised to the largest exponent of one. Let's see a second-degree polynomial:

x23x+7\footnotesize x^2-3x+7

This is an example of trinomial, as there are three terms. We can, of course, have second-degree binomials: in these polynomials, only two of the terms appear, one of them necessarily being the one with x2x^2 (e.g., 3x243x^2 - 4 or 2x2x2x^2 - x).

🙋 If there are two or more indeterminates in your polynomial, its degree corresponds to the maximum value of the sum of the exponents among all terms. For example, the polynomial 3xy2+4x23y3xy^2+4x^2-3y has degree 33 as the exponent of xx in 3xy23xy^2 is 11 and the exponent of yy is 22.

For specific combinations of indeterminates and coefficients, we can reduce the polynomial — we can write it in the product of two polynomials, which, in turn, can't be reduced further. When dealing with factorization, we assume that the coefficients of the original and reduced polynomials are integers. This allows us to define specific rules for the process.

This process (also known as factorization of the polynomial) is a fundamental tool in mathematics. It allows us to simplify expressions, often finding patterns previously hidden, and, in some cases, it may help us solve equations. In the next section, you'll learn a method to find the factorization of a trinomial. Are you ready to learn how to calculate the reverse FOIL method?

🙋 We talked in detail about this process at our factoring trinomials calculator. Give it a try if you are interested!

What is the reverse FOIL method?

The reverse FOIL method is a factorization algorithm we can apply to trinomials of second degree in one indeterminate to find two irreducible binomials. The name FOIL is an acronym, where:

  • F stands for first;
  • O stands for outer;
  • I stands for inner; and
  • L stands for last.

The algorithm involves finding the first and last terms of the factored binomials and then calculating the products of the inner and outer pairs: using a trial-by-error approach, we compute the sum of these products until it matches the second term of our original trinomial.

The reverse FOIL method returns a product of two first-degree binomials. This factorization is the fastest way to solve a quadratic equation. If you equate the product to zero, the solution of each binomial is a solution of the original quadratic equation! It may be even faster than our quadratic formula calculator.

How to do the reverse FOIL on a generic polynomial? Jump to the next section!

🙋 If there is a reverse FOIL, there should also be a "direct" FOIL, right? Yes! Visit our FOIL calculator to learn the opposite operation of factorization and a simple way to multiply binomials!

How to calculate the reverse FOIL method step-by-step

We introduced you to the reverse FOIL: how to calculate it, though? Take a generic second-degree trinomial in one indeterminate:

ax2+bx+c\footnotesize ax^2+bx+c

Above, aa, bb, and cc are the coefficients. Our purpose is to find a couple of binomials (αx+β)(\alpha x+\beta) and (γx+δ)(\gamma x + \delta) such that:

(αx+β)(γx+δ)=ax2+bx+c\footnotesize (\alpha x+\beta)(\gamma x + \delta)=ax^2+bx+c

To do so, we begin by finding two values for α\alpha and γ\gamma such that:

αγ=a\footnotesize \alpha\cdot\gamma=a

You can see how we used the first coefficients of the binomial, hence the F of FOIL!

⚠️ Notice that there may be more than one possible combination of α\alpha and γ\gamma. We will deal with this later!

Next up: find the possible factorizations of cc. To do so, we will jump to the end of FOIL and use the last coefficients:

βδ=c\footnotesize \beta\cdot\delta = c

For β\beta and δ\delta, too, we can find multiple combinations.

The next step in calculating the reverse FOIL method involves computing the products of the outer and inner coefficients. Why? Let's compute the product of the binomials:

(αx+β)(γx+δ)= αγx2+αδx +βγx+βδ\footnotesize \begin{split} (\alpha x+\beta)(\gamma x + \delta)=\ &\alpha\gamma x^2 + \alpha\delta x\\\ &+\beta\gamma x + \beta\delta \end{split}

Here you can see how the first and last terms of the sum correspond with the coefficients aa and cc as we've calculated earlier. However, there are two terms that multiply with xx. So, to find bb, we need to compute:

αδ+βγ=b\footnotesize \alpha\delta+\beta\gamma = b

That's it — in theory!

In practice, we need a bit of guesswork. If you want to calculate the reverse FOIL method, follow these steps:

  1. Find a pair of numbers α\alpha and γ\gamma that satisfy αγ=a\alpha\cdot\gamma = a.

  2. Find all possible pairs of numbers β\beta and δ\delta such that βδ=c\beta\cdot\delta=c.

  3. Compute the products αδ\alpha\cdot\delta (the product of the outer coefficients) and γδ\gamma\cdot\delta (the product of the inner coefficients) for all possible pairs of β\beta and δ\delta.

  4. Compute the sum of a pair of products you found in the step above.

    • If the result equals bb (αδ+βγ=b\alpha\cdot\delta+\beta\cdot\gamma = b), you found a factorization of the trinomial. Write the result as (αx+β)(γx+δ)(\alpha x+\beta)(\gamma x + \delta).

    • If the previous step didn't return a valid factorization, repeat it for the next pair of products you found in step 3.

  5. It may happen that no products allow you to find a valid factorization. If this is the case, return to step 1, find another possible combination of α\alpha and γ\gamma, and repeat the algorithm's steps from there.

  6. If, even after exhausting all possible factorizations of aa, you didn't find a valid combination of inner and outer products, it means you stumbled upon an irreducible polynomial!

Reverse FOIL: how to calculate in an example

Now that you know what the reverse FOIL is and how to calculate it, we can try to apply it to a polynomial with numerical coefficients! Let's take the trinomial 6x27x56x^2-7x-5.

Let's apply the reverse FOIL method.

  1. Find a pair of numbers that, multiplied, returns 66 (the first coefficient, aa): 11 and 66 are a valid choices.

  2. Move onto the last coefficient: 5-5 (the sign matters): in this case, we have two possible choices: 1-1 and 55, and 11 and 5-5.

  3. Let's consider the first pair. With such combinations, we'd create the factorization (x1)(6x+5)(x-1)(6x+5). Let's compute the products of the inner and outer coefficients: 16=6-1\cdot 6 = -6 and 15=51\cdot 5 = 5.

  4. Calculate the sum of these results: 6+5=1-6+5=-1. It doesn't match the second coefficient. Hence this is not a valid factorization.

  5. Repeat steps 3 and 4 for the other pair of coefficients: the products will be 16=61\cdot6=6 and 15=51\cdot -5 = -5, which sum to 65=16-5=1: again, a nonvalid factorization. Is this it? Nope!

  6. Change the choice of factors for the first coefficient: this time, we will use 22 and 33.

  7. Let's consider the same pair of numbers that factorize the last coefficient:

    • For the first pair, we'd create the factorization (2x1)(3x+5)(2x-1)(3x+5). Compute the inner and outer products: 13=3-1\cdot3=-3 and 25=102\cdot5=10. Calculate their sum: 3+10=7-3+10=7. Still not valid, but we're just a sign away. Maybe in the next step?

    • The second pair would give us the factorization (2x+1)(3x5)(2x+1)(3x-5), with inner and outer products 13=31\cdot3=3 and 25=102\cdot-5=-10. Their sum is 310=73-10=-7, which matches the second coefficient of the original trinomial!

At last, we found a valid factorization of the original trinomial:

6x27x5=(2x+1)(3x5)\footnotesize 6x^2-7x-5 = (2x+1)(3x-5)

Is this it? Nope! Don't limit yourself, and take a peek at negative numbers. Follow the previous steps with the possible combinations 6=166=-1\cdot-6 and 6=236=-2\cdot-3. You may even see the result already: adding a minus sign on both numbers won't affect the process. We can find a further factorization:

6x27x5=(2x1)(3x+5)\footnotesize 6x^2-7x-5 = (-2x-1)(-3x+5)

Generally speaking, if you find a factorization, there will be an "evil twin" with some signs changed in the final binomials.

🙋 The prime factorization calculator may help you with the task of finding the factors of the first and last coefficients of the original trinomial!

How to use our reverse FOIL calculator

You just learned how to reverse FOIL a trinomial by hand, so learning how to use our reverse FOIL calculator will be as easy as pie.

Start by inserting the coefficients of the trinomial. You can insert positive numbers as well as negative ones. If you can compute a factorization, we will print the result in the blink of an eye. You can even choose to see the steps we used to find the result!

If you don't insert all the coefficients, we won't compute the reverse FOIL method. You can't apply it to binomials, but we will compute the factorization if possible!

FAQ

What is the reverse FOIL method?

The reverse FOIL method is a factorization algorithm for second-degree trinomials in a single indeterminate that allows you to find the pair of irreducible binomials that returns the original trinomial when multiplied. The reverse FOIL method involves moderate guesswork but is relatively quick in general.

How do you calculate the reverse FOIL method?

To calculate the reverse FOIL method on a generic second-degree trinomial ax2 + bx + c, follow these easy steps:

  1. Find a pair of numbers α and γ that satisfies α × γ = a.
  2. Find all pairs of numbers ß and δ that satisfy ß × δ = c.
  3. Write the four numbers as the product of two binomials (αx + ß)(γx + δ).
  4. Compute the products of the inner and outer coefficients and the sum of these products: ßγ + αδ.
  5. If the sum equals b, (αx + ß)(γx + δ) is a valid factorization.

How do you factorize x² + 4x + 3?

You can factorize x2 + 4x + 3 using the reverse FOIL method. To do so, follow these simple steps:

  1. Find the factors of the first coefficient. In this case, 1 = 1 × 1.
  2. Find the possible combinations of factors of 3: 1 and 3, and 3 and 1.
  3. Create the two possible products of binomials, and compute the products of the inner and outer terms. Sum these products, and check if the result matches the second coefficient (4):
    • (x + 1)(x + 3), with sum of products 3 + 1 = 4

The factorization of x2 + 4x + 3 is then (x + 1)(x + 3).

How do you solve a quadratic equation with factorization?

You can use the reverse FOIL method. If none of the coefficients are zero, you can find a combination of the factors of coefficients a and c as the first and last terms of a pair of binomials. If the sum of the products of the inner and outer terms matches the second coefficient, you found a factorization of the equation. Then solve the two binomials.

Davide Borchia
First coefficient
Second coefficient
x
Third coefficient
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