Lesson The difference of squares formula

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The difference of squares formula

The difference of squares formula is

a%5E2+-+b%5E2+=+%28a%2Bb%29%2A%28a-b%29

.

The formula is valid for any real numbers

and

.
In particular, it is valid for all integer numbers.
The formula is valid for the complex numbers too.

The proof of the formula is very simple. It follows straightforward from the direct calculations:

%28a+%2B+b%29%2A%28a-b%29+=+a%5E2+%2B+b%2Aa+-+a%2Ab+-+b%5E2+=+a%5E2+-+b%5E2

%28a+%2B+b%29%2A%28a-b%29+=+a%5E2+%2B+b%2Aa+-+a%2Ab+-+b%5E2+=+a%5E2+-+b%5E2

.

As you see, the distributive and the commutative properties of addition and multiplication operations over the real numbers are used in derivation the formula.

The difference of squares formula is useful in a number of applications. You should memorize it.
In some cases, this formula can facilitate calculations and make them mentally without using paper and pencil (or calculator).

Example 1

Calculate

.

Solution
99%5E2-1+=+%2899%2B1%29%2A%2899-1%29+=+100%2A98+=+9800

Example 2

Calculate

.

Solution
999%5E2-1+=+%28999%2B1%29%2A%28999-1%29+=+1000%2A998+=+998000

Example 3

Calculate

.

Solution
49%5E2-1+=+%2849%2B1%29%2A%2849-1%29+=+50%2A48+=+2400

Do yourself

Apply the difference of squares formula to check that

51%5E2-1+=+2600

.

The difference of squares formula is applicable not only to numbers. It is applicable for binomials too. For example,

x%5E2+-+1+=+%28x+%2B+1%29%2A%28x-1%29

.

You can check validity of the last formula directly by performing all relevant calculations: opening the brackets, multiplying the terms and combining the like terms.
You will get the same result. It is not surprising, because addition and multiplication operations over polynomials have the same distributive and commutative properties
as over the real numbers. Thus, the difference of squares formula is simply the useful shortcut formula.
It may help you when you need to factor polynomials.

Example 4

Factor the binomial 4x%5E2-9

.

Solution
4x%5E2-9+=+%28%282x%29%5E2-3%29%2A%28%282x%29%5E2%2B3%29+=+%282x%2B3%29%2A%282x-1%29

Example 5

Factor the binomial 49x%5E2-81

.

Solution
49x%5E2-81+=+%28%287x%29%5E2-9%29%2A%28%287x%29%5E2%2B9%29+=+%287x%2B9%29%2A%287x-9%29

.

Note. Pay attention how the brackets are used to group monomials according the pattern of the difference of squares formula.

Example 6

Factor the binomial x%5E4-1

.

Solution
x%5E4-1+=+%28x%5E2-1%29%2A%28x%5E2%2B1%29+=+%28x-1%29%2A%28x%2B1%29%2A%28x%5E2%2B1%29

Example 7

Factor the binomial a%5E4-b%5E4

.

Solution

.

You got the decomposition of the bi-quadratic binomial a%5E4-b%5E4

into the product of the two linear binomials a-b

and

and one quadratic binomial a%5E2%2Bb%5E2

Note 1. You can get another decomposition of the binomial by rolling up the factors and of the previous factorization into the product
            .

            Then you get
            .

Note 2. The quadratic binomial can not be factored into the product of linear binomials on and over the real numbers.
            For those acquainted with the complex numbers, this quadratic binomial can be factored over the complex numbers:
            ,
            where is the imaginary complex numbers unit. Indeed,
            .
            But any factorization of the binomial into the product of linear binomials on and is not possible over the real numbers.

Note 3. In contrast to the quadratic binomial , the bi-quadratic binomial can be factored into the product of two polynomials of lesser degree.
            Some trick is used: add and distract the monomial to/from the binomial , then apply the shortcut square of the sum formula and
            the difference of squares formula. You get

            .

Example 8

Factor the expression a%5E2%2B2ab%2Bb%5E2-c%5E2

.

Solution
First, apply the square of the sum formula and then apply the difference of squares formula. You get

a%5E2%2B2ab%2Bb%5E2-c%5E2+=+%28a%2Bb%29%5E2-c%5E2+=+%28a%2Bb%2Bc%29%2A%28a%2Bb-c%29

a%5E2%2B2ab%2Bb%5E2-c%5E2+=+%28a%2Bb%29%5E2-c%5E2+=+%28a%2Bb%2Bc%29%2A%28a%2Bb-c%29

Summary

The difference of squares formula

a%5E2+-+b%5E2+=+%28a%2Bb%29%2A%28a-b%29

is useful shortcut multiplication formula.
At the same time, you can use it to factor binomials when applicable.

For similar lessons see
    The square of the sum formula and
    The square of the difference formula
under the current topic in this site.

The difference of squares formula is widely used in rationalizing fractions by making their denominator free of square roots. For details and examples see the lesson
    HOW TO rationalize a fraction by making its denominator free of square roots
under the current topic in this site. It is used also in simplifying rational expressions. For details and examples see the lesson
    Simplifying rational expressions with the use the shortcut multiplication formulas
under the current topic in this site.

For the list of all shortcut quadratic multiplication formulas see the lesson
    OVERVIEW of shortcut quadratic multiplication formulas
under the current topic in this site.

For factoring the binomials of the third degree a%5E3-b%5E3

see the lessons
The difference of cubes formula,
The sum of cubes formula
under the current topic in this site.

For factoring the binomials of high degrees x%5En-a%5En

see the lessons
Factoring the binomials x%5En-a%5En

,
Factoring the binomials x%5En%2Ba%5En

for odd degrees,
OVERVIEW of Factoring the binomials x%5En-a%5En

and

under the current topic in this site.

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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