How to factor these 3 questions?

Question

Hi , I need help factoring these , if possible with steps so I can better understand. 

 

1.  a^3(a+b)^2 + b^3(a+b)^2

 

2. 8x^4(x-4) - 27x(x-4)

 

3. (x+1)^2 -(x+1) -6

Answer 1

There are two ways to factor.

 

a) Extracting the greatest common factor (GCF).

b) Using FOIL

 

First, you need to examine the expression and see which method to apply.  I would start by looking for the GCF.

 

 

1)

 

Now that the GCF is (a + b)².  So we can factor the expression as

 

(a + b)²(a³ + b³)

 

 

2)

 

The GCF is x(x - 4).  So we can factor the expression as

 

x(x - 4)(8x³ - 27)

 

 

Do the same for the last one.

Answer 2

Hi Alex.

 

Let's see what we can do.

 

1)

a³(a+b)² + b³(a+b)²

We can factor out an a+b)² out of each term.  This gives us

(a+b)²(a³+b³)

 

Now recall how to factor the sum of two cubes...

a3 + b3 = (a+b)(a²-ab+b²)

 

So now we have

(a+b)² (a+b) (a²-ab+b²)

(a+b)³(a²-ab+b²)

 

 

2)8x⁴(x-4) - 27x(x-4)

First let's recognize that 8 = 23 and 27 = 3³

 

We can factor out an x(x-4) from each term

x(x-4) [(2x)³ - 3³]

 

Notice we are left with the difference of two cubes in [ ]

to factor the difference of two cubes

a³ - b³ = (a-b)(a²+ab+b²)

 

x(x-4){(2x-3)[(2x)²+(2x)(3)+3²]}

x(x-4)(2x-3)(4x²+6x+9)

 

And finally

 

3) (x+1)² -(x+1) -6

If we foil the first term and distribute the -1 to the second term, we have

x²+2x+1-x-1-6

Combining like terms

x² + x - 6

which can be factored as below

(x+3)(x-2)