Example: 3x2 + 9x3 + 12x4 factored into 3x2(1 + 3x + 4x2) why is the 1 there?

Question

i get that 3x² + 9x³ + 12x⁴ factors into 3x^{2( ) and} 9x³ goes into 3x and 12x⁴ goes into 4x² but why is the 1 there it seems really random.

 

Answer 1

Working backwards from your answer, 3x²(1 + 3x + 4x²),

 

go ahead and distribute every term in the parentheses by 3x² and you will get your original equation of

 

3x² + 9x³ + 12x⁴. 

 

 

The explanations listed are correct; from the original equation of 3x² + 9x³ + 12x⁴,  when you divide 3x² by itself, you are left with a factor of 1; hence, the reason for the 1 in the factored form of this equation:  (1 + 3x + 4x²)3x² .

Answer 2

Quite simply... when you divide the 3x² out you do it from every term. 

The first term left inside the parentheses will be (3x²/3x²) = 1

 

This is no different than how you got the other terms in the parentheses

9x³/3x² = 3x

12x⁴/3x² = 4x²

 

Thus 3x² + 9x³ + 12x⁴ = 3x²(1 + 3x + 4x²)

Answer 3

If you multiply 3x²(1 + 3x + 4x²) out, you get:

 

3x²*1 + 3x²*3x + 3x²*4x² = 3x² + 9x³ + 12x⁴

 

which is what you started with.  Now try it without the 1:

 

3x²(3x + 4x²) = 9x² + 12x⁴

 

which is wrong - you're missing the 3x² term.  The one holds the place of the 3x² term once you factor it out.