In numerous ways, factoring is around patterns—if you identify the patterns that number make as soon as they room multiplied together, you can use those fads to separate these numbers right into their separation, personal, instance factors.

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Some exciting patterns arise once you are working through cubed quantities within polynomials. Special, there space two an ext special cases to consider: a3 + b3 and also a3 – b3.

Let’s take a look at at just how to element sums and also differences of cubes.


The hatchet “cubed” is offered to describe a number elevated to the 3rd power. In geometry, a cube is a six-sided shape with same width, length, and height; due to the fact that all these actions are equal, the volume the a cube v width x have the right to be stood for by x3. (Notice the exponent!)

Cubed number get big very quickly. 13 = 1, 23 = 8, 33 = 27, 43 = 64, and 53 = 125.

Before looking in ~ factoring a amount of 2 cubes, let’s look at the possible factors.

It transforms out that a3 + b3 have the right to actually it is in factored together (a + b)(a2 – abdominal muscle + b2). Let’s inspect these determinants by multiplying.


does (a + b)(a2 ab + b2) = a3 + b3?

(a)(a2 – ab + b2) + (b)(a2 – ab +b2)

Apply the distributive property.

(a3 – a2b + ab2) + (b)(a2 - abdominal muscle + b2)

Multiply by a.

(a3 – a2b + ab2) + (a2b – ab2 + b3)

Multiply by b.

a3 – a2b + a2b + ab2 – ab2 + b3

Rearrange terms in stimulate to combine the choose terms.

a3 + b3

Simplify


Did you view that? 4 of the state cancelled out, leaving us with the (seemingly) straightforward binomial a3 + b3. So, the factors are correct.

You can use this pattern to variable binomials in the kind a3 + b3, otherwise known as “the sum of cubes.”

The amount of Cubes

A binomial in the kind a3 + b3 deserve to be factored as (a + b)(a2 – abdominal muscle + b2).

Examples:

The factored type of x3 + 64 is (x + 4)(x2 – 4x + 16).

The factored form of 8x3 + y3 is (2x + y)(4x2 – 2xy + y2).


Example

Problem

Factor x3 + 8y3.

x3 + 8y3

Identify the this binomial fits the sum of cubes pattern: a3 + b3.

a = x, and b = 2y (since 2y • 2y • 2y = 8y3).

(x + 2y)(x2 – x(2y) + (2y)2)

Factor the binomial as

(a + b)(a2 – ab + b2), substituting a = x and b = 2y right into the expression.

(x + 2y)(x2 – x(2y) + 4y2)

Square (2y)2 = 4y2.

Answer

(x + 2y)(x2 – 2xy + 4y2)

Multiply −x(2y) = −2xy (writing the coefficient first.


And that’s it. The binomial x3 + 8y3 have the right to be factored as (x + 2y)(x2 – 2xy + 4y2)! Let’s try another one.

You should constantly look for a typical factor before you follow any of the fads for factoring.


Example

Problem

Factor 16m3 + 54n3.

16m3 + 54n3

Factor the end the usual factor 2.

2(8m3 + 27n3)

8m3 and also 27n3 are cubes, so friend can factor 8m3 + 27n3 together the sum of 2 cubes: a = 2m, and b = 3n.

2(2m + 3n)<(2m)2 – (2m)(3n) + (3n)2>

Factor the binomial 8m3 + 27n3 substituting a = 2m and b = 3n into the expression (a + b)(a2 – ab + b2).

2(2m + 3n)<4m2 – (2m)(3n) + 9n2>

Square: (2m)2 = 4m2 and also (3n)2 = 9n2.

Answer

2(2m + 3n)(4m2 – 6mn + 9n2)

Multiply −(2m)(3n) = −6mn.


Factor 125x3 + 64.

A) (5x + 64)(25x2 – 125x + 16)

B) (5x + 4)(25x2 – 20x + 16)

C) (x + 4)(x2 – 2x + 16)

D) (5x + 4)(25x2 + 20x – 64)


Show/Hide Answer

A) (5x + 64)(25x2 – 125x + 16)

Incorrect. Inspect your worths for a and also b here. B3 = 64, so what is b? The exactly answer is (5x + 4)(25x2 – 20x + 16).

B) (5x + 4)(25x2 – 20x + 16)

Correct. 5x is the cube source of 125x3, and also 4 is the cube source of 64. Substituting these worths for a and b, you find (5x + 4)(25x2 – 20x + 16).

C) (x + 4)(x2 – 2x + 16)

Incorrect. Check your worths for a and also b here. A3 = 125x3, for this reason what is a? The exactly answer is (5x + 4)(25x2 – 20x + 16).

D) (5x + 4)(25x2 + 20x – 64)

Incorrect. Inspect the math signs; the b2 term is positive, no negative, as soon as factoring a sum of cubes. The exactly answer is (5x + 4)(25x2 – 20x + 16).

Difference of Cubes


Having seen how binomials in the kind a3 + b3 can be factored, it must not come as a surprise that binomials in the type a3 – b3 deserve to be factored in a similar way.

The difference of Cubes

A binomial in the kind a3 – b3 can be factored as (a – b)(a2 + abdominal + b2).

Examples:

The factored kind of x3 – 64 is (x – 4)(x2 + 4x + 16).

The factored kind of 27x3 – 8y3 is (3x – 2y)(9x2 + 6xy + 4y2).

Notice the the straightforward construction the the factorization is the exact same as it is for the sum of cubes; the difference is in the + and also – signs. Take a moment to to compare the factored type of a3 + b3 v the factored form of a3 – b3.


Factored kind of a3 + b3:

(a + b)(a2 – abdominal muscle + b2)

Factored form of a3 – b3:

(a – b)(a2 + ab + b2)


This have the right to be tricky to remember because of the various signs—the factored kind of a3 + b3 contains a negative, and also the factored kind of a3 – b3 consists of a positive! Some civilization remember the various forms favor this:

“Remember one sequence of variables: a3 b3 = (a b)(a2 abdominal muscle b2). There room 4 absent signs. Whatever the first sign is, that is likewise the second sign. The 3rd sign is the opposite, and also the 4th sign is constantly +.”

Try this for yourself. If the an initial sign is +, as in a3 + b3, follow to this strategy how do you to fill in the rest: (a b)(a2 abdominal muscle b2)? walk this technique help you remember the factored form of a3 + b3 and a3 – b3?

Let’s go ahead and look at a couple of examples. Psychic to element out all common factors first.


Example

Problem

Factor 8x3 – 1,000.

8(x3 – 125)

Factor out 8.

8(x3 – 125)

Identify that the binomial fits the pattern a3 - b3: a = x, and also b = 5 (since 53 = 125).

8(x - 5)

Factor x3 – 125 as (a – b)(a2 + abdominal muscle + b2), substituting a = x and also b = 5 right into the expression.

8(x – 5)(x2 + 5x + 25)

Square the very first and last terms, and rewrite (x)(5) together 5x.

Answer

8(x – 5)(x2 + 5x + 25)


Let’s see what happens if girlfriend don’t variable out the typical factor first. In this example, it deserve to still be factored as the difference of 2 cubes. However, the factored kind still has usual factors, which must be factored out.


Example

Problem

Factor 8x3 – 1,000.

8x3 – 1,000

Identify the this binomial fits the pattern a3 - b3: a = 2x, and also b = 10 (since 103 = 1,000).

(2x – 10)<(2x)2 + 2x(10) + 102>

Factor together (a – b)(a2 + ab + b2), substituting a = 2x and b = 10 into the expression.

(2x – 10)(4x2 + 20x + 100)

Square and also multiply: (2x)2 = 4x2,

(2x)(10) = 20x, and also 102 = 100.

2(x – 5)(4)(x2 + 5x + 25)

Factor the end remaining common factors in every factor. Element out 2 from the an initial factor, factor out 4 native the 2nd factor.

(2 • 4)(x – 5)(x2 + 5x + 25)

Multiply the number factors.

Answer

8(x – 5)(x2 + 5x + 25)


As you have the right to see, this last instance still worked, however required a couple of extra steps. That is always a great idea to factor out all common factors first. In some cases, the just efficient method to factor the binomial is to aspect out the typical factors first.

Here is one much more example. Keep in mind that r9 = (r3)3 and also that 8s6 = (2s2)3.


Example

Problem

Factor r9 – 8s6.

r9 – 8s6

Identify this binomial as the difference of 2 cubes. As presented above, the is. Utilizing the laws of exponents, rewrite r9 together (r3)3.

(r3)3 – (2s2)3

Rewrite r9 as (r3)3 and rewrite 8s6 together (2s2)3.

Now the binomial is written in terms of cubed quantities. Reasoning of a3 – b3,

a = r3 and b = 2s2.

(r3 – 2s2)<(r3)2 + (r3)(2s2) + (2s2)2>

Factor the binomial as

 (a – b)(a2 + ab + b2), substituting a = r3 and also b = 2s2 right into the expression.

(r3 – 2s2)(r6 + 2 r3s2+ 4s4)

Multiply and square the terms.

Answer

(r3 – 2s2)(r6 + 2r3s2 + 4s4)


Using the difference of cubes, identify the product the 3(x – 3y)(x2 + 3xy + 9y2).

A) x3 – y3

B) 3x – 81y

C) 3x3 + 81y3

D) 3x3 – 81y3


Show/Hide Answer

A) x3 – y3

Incorrect. If this were true, the expression shown over would it is in (x – y)(x2 + xy + y2). The correct answer is 3x3 – 81y3.

B) 3x – 81y

Incorrect. Neither of the state in this binomial is a cubed number! The exactly answer is

3x3 – 81y3.

C) 3x3 + 81y3

Incorrect. Examine your signs. If this expression drops into the distinction of cubes category, the symbol between 3x3 and 81y3 should be –. The correct answer is 3x3 – 81y3.

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D) 3x3 – 81y3

Correct. Recognizing that this expression is in the type (a – b)(a2 + ab + b2), you uncover a = x and also b = 3y. This means that the result a3 – b3 monomial is x3 – 27y3. It also needs come be multiplied by the coefficient 3: 3x3 – 81y3.

Summary


You encounter some interesting patterns as soon as factoring. 2 special cases—the sum of cubes and also the difference of cubes—can assist you factor some binomials that have a level of 3 (or higher, in some cases). The special instances are:

A binomial in the kind a3 + b3 deserve to be factored as (a + b)(a2 – abdominal + b2) A binomial in the kind a3 – b3 have the right to be factored as (a – b)(a2 + abdominal muscle + b2)