Have you ever heard your teacher or fellow students talking about the FOIL method? They're probably not talking about the type of foil you use for fencing or in the kitchen. Instead, the FOIL method stands for "first, outer, inner, last," a mnemonic or memory device that helps you remember how to multiply two binomials together, which is exactly what you're doing when you take the square of a binomial.
TL;DR (Too Long; Didn't Read)
To square a binomial, write out the multiplication and use the FOIL method to add the sums of the first, outer, inner and last terms. The result is the square of the binomial.
A Quick Refresher on Squaring
Before you go any further, take a second to refresh your memory on what it means to square a number, regardless of whether it's a variable, a constant, a polynomial (which includes binomials) or anything else. When you square a number, you multiply it by itself. So if you square x, you have x × x, which can also be written as x2. If you square a binomial like x + 4, you have (x + 4)2 or once you write out the multiplication, (x + 4) × (x + 4). With that in mind, you're ready to apply the FOIL method to squaring binomials.
Write Out the Multiplication
Write out the multiplication implied by the squaring operation. So if your original problem is to evaluated (y + 8)2, you'd write it as:
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(y + 8)(y + 8)
Apply the FOIL Method
Apply the FOIL method starting with the "F," which stands for the first terms of each polynomial. In this case the first terms are both y, so when you multiply them together you have:
Next, multiply the "O" or outer terms of each binomial together. That's the y from the first binomial and the 8 from the second binomial, since they're on the outer edges of the multiplication you wrote out. That leaves you with:
The next letter in FOIL is "I," so you'll multiply the inner terms of the polynomials together. That's the 8 from the first binomial and the y from the second binomial, giving you:
(Note that if you're squaring a polynomial, the "O" and "I" terms of FOIL will always be the same.)
The last letter in FOIL is "L," which stands for multiplying the last terms of the binomials together. That's the 8 from the first binomial and the 8 from the second binomial, which gives you:
8 × 8 = 64
Add the FOIL Terms Together
Add the FOIL terms you just calculated together; the result will be the square of the binomial. In this case the terms were y2, 8_y_, 8_y_ and 64, so you have:
y2 + 8_y_ + 8_y_ + 64
You can simplify the result by adding both 8_y_ terms, which leaves you with the final answer:
y2 + 16_y_ + 64
The FOIL is a quick, easy way of remembering how to multiply binomials. But it only works for binomials. If you're dealing with polynomials that have more than two terms, you'll have to apply the distributive property.