In numerous ways, factoring is around patterns—if you identify the trends that number make once they room multiplied together, you deserve to use those fads to different these numbers into their separation, personal, instance factors.

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Some interesting patterns arise when you are working with cubed quantities within polynomials. Specifics there are two more special situations to consider: a3 + b3 and also a3 – b3.

Let’s take it a watch at exactly how to aspect sums and also differences the cubes.


The hatchet “cubed” is used to explain a number elevated to the third power. In geometry, a cube is a six-sided shape with same width, length, and height; since all these measures are equal, the volume that a cube v width x deserve to be stood for by x3. (Notice the exponent!)

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

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

It transforms out the a3 + b3 deserve to actually be factored as (a + b)(a2 – ab + b2). Let’s examine these determinants by multiplying.


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

(a)(a2 – abdominal muscle + b2) + (b)(a2 – ab +b2)

Apply the distributive property.

(a3 – a2b + ab2) + (b)(a2 - ab + b2)

Multiply by a.

(a3 – a2b + ab2) + (a2b – ab2 + b3)

Multiply by b.

a3 – a2b + a2b + ab2 – ab2 + b3

Rearrange terms in order to incorporate the like terms.

a3 + b3

Simplify


Did you check out that? four of the terms cancelled out, leaving us v the (seemingly) an easy binomial a3 + b3. So, the components are correct.

You deserve to use this sample to factor binomials in the form a3 + b3, otherwise well-known as “the amount of cubes.”

The sum of Cubes

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

Examples:

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

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


Example

Problem

Factor 16m3 + 54n3.

16m3 + 54n3

Factor out the common factor 2.

2(8m3 + 27n3)

8m3 and 27n3 space cubes, so friend can variable 8m3 + 27n3 as the sum of two cubes: a = 2m, and b = 3n.

2(2m + 3n)<(2m)2 – (2m)(3n) + (3n)2>

Factor the binomial 8m3 + 27n3 substituting a = 2m and also b = 3n right into the expression (a + b)(a2 – abdominal + 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. Check your values for a and also b here. B3 = 64, for this reason what is b? The correct answer is (5x + 4)(25x2 – 20x + 16).

B) (5x + 4)(25x2 – 20x + 16)

Correct. 5x is the cube root of 125x3, and 4 is the cube root 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 values for a and b here. A3 = 125x3, therefore what is a? The exactly answer is (5x + 4)(25x2 – 20x + 16).

D) (5x + 4)(25x2 + 20x – 64)

Incorrect. Check the math signs; the b2 term is positive, not negative, once factoring a sum of cubes. The exactly answer is (5x + 4)(25x2 – 20x + 16).

Difference the Cubes


Having seen just how binomials in the kind a3 + b3 deserve to be factored, it need to not come as a surprise that binomials in the kind a3 – b3 have the right to be factored in a similar way.

The difference of Cubes

A binomial in the form a3 – b3 can be factored together (a – b)(a2 + ab + b2).

Examples:

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

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

Notice the the an easy construction of the factorization is the exact same as that is for the sum of cubes; the difference is in the + and also – signs. Take it a minute to compare the factored type of a3 + b3 through the factored kind of a3 – b3.


Factored kind of a3 + b3:

(a + b)(a2 – abdominal + b2)

Factored form of a3 – b3:

(a – b)(a2 + abdominal muscle + b2)


This have the right to be tricky to remember due to the fact that of the different signs—the factored kind of a3 + b3 contains a negative, and the factored form of a3 – b3 has a positive! Some civilization remember the different forms prefer this:

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

Try this for yourself. If the very first sign is +, together in a3 + b3, according to this strategy how do you fill in the rest: (a b)(a2 ab b2)? go this method help friend remember the factored type of a3 + b3 and a3 – b3?

Let’s walk ahead and look in ~ a pair of examples. Remember to variable out all typical factors first.


Example

Problem

Factor 8x3 – 1,000.

8(x3 – 125)

Factor out 8.

8(x3 – 125)

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

8(x - 5)

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

8(x – 5)(x2 + 5x + 25)

Square the an initial 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 usual factor first. In this example, it deserve to still be factored together the distinction of two cubes. However, the factored type still has common factors, which have to be factored out.


Example

Problem

Factor 8x3 – 1,000.

8x3 – 1,000

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

(2x – 10)<(2x)2 + 2x(10) + 102>

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

(2x – 10)(4x2 + 20x + 100)

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

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

2(x – 5)(4)(x2 + 5x + 25)

Factor out remaining typical factors in every factor. Aspect out 2 from the an initial factor, variable out 4 from the second factor.

(2 • 4)(x – 5)(x2 + 5x + 25)

Multiply the numerical factors.

Answer

8(x – 5)(x2 + 5x + 25)


As you can see, this last instance still worked, however required a pair of extra steps. It is constantly a good idea to element out all common factors first. In some cases, the only efficient means to factor the binomial is to element out the typical factors first.

Here is one more example. Note that r9 = (r3)3 and that 8s6 = (2s2)3.


Example

Problem

Factor r9 – 8s6.

r9 – 8s6

Identify this binomial together the distinction of two cubes. As displayed above, it is. Making use of 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 also b = 2s2.

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

Factor the binomial as

 (a – b)(a2 + abdominal + b2), substituting a = r3 and 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, determine the product of 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 be (x – y)(x2 + xy + y2). The exactly answer is 3x3 – 81y3.

B) 3x – 81y

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

3x3 – 81y3.

C) 3x3 + 81y3

Incorrect. Check your signs. If this expression falls into the distinction of cubes category, the symbol in 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 form (a – b)(a2 + abdominal + b2), you uncover a = x and b = 3y. This means that the result a3 – b3 monomial is x3 – 27y3. It likewise needs come be multiply by the coefficient 3: 3x3 – 81y3.

Summary


You encounter some exciting patterns once factoring. 2 special cases—the sum of cubes and also the distinction of cubes—can assist you aspect some binomials that have a degree of three (or higher, in part cases). The special cases are:

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