Factor: 3y^{2n+1} + 12y^{3n+2}

Question

Factor: 3y²ⁿ⁺¹ + 12y³ⁿ⁺²

 

 

Answer 1

3y²ⁿ⁺¹ + 12y³ⁿ⁺²

 

You could start by using the rules of exponents to write each term as a product to make it easier to work with, remember for instance that  y²ⁿ⁺¹= y²ⁿy. Also, y³ⁿ⁺²= y³ⁿy^2. This allows expression to be rewritten as: 

 

3y²ⁿy + 12y³ⁿy²

 

The second term can be expressed differently as well so you can see the common factor between the terms more clearly, 

 

3y²ⁿy + 12y²ⁿyⁿy²

 

Factor out 3y²ⁿ,

 

=3y²ⁿ(y + 4yⁿy²)

 

This can be factored further by pulling out y, 

 

=3y²ⁿy(1 + 4yⁿy)

 

Now use the rules of exponents again to clean up a bit, 

 

= 3y²ⁿ⁺¹(1+4yⁿ⁺¹)

 

 

Also, you could have just noticed from the start that there must be a way to factor out 3y²ⁿ⁺¹ and brute forced it by saying, 

 

3y²ⁿ⁺¹ + 12y³ⁿ⁺² = 3y²ⁿ⁺¹(1+4y^something)

 

And then deducing what that exponent would have to be. 

 

 

 

 

Answer 2

You can pull out a 3 from both and a y²ⁿ⁺¹

 

So you have 3y²ⁿ⁺¹ on the outside

 

so you are left with

3y²ⁿ⁺¹(1+4yⁿ⁺¹)