Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation.

Question

zehanz · Answer

To factor the first one, look for two numbers, with sum b and product c. Once you've found them (provided they are easy enough to figure out), you can write the trinomial as follows (suppose the numbers you've found are called p and q):$$x^2+bx+c=(x+p)(x+q)$$

anonymous · Answer

is there more then one way to factor this?

zehanz · Answer

No. According to the Fundamental Theorem of Algebra, polynomials can be written as the product of 1st degree polynomials in a unique way!

anonymous · Answer

thank you

zehanz · Answer

As for the second form, I mostly cannot quickly find a way to factor it. To understand why, let me give an example of an already factored form:$$(2x-3)(x+4)=2x^2-3x+8x-12=2x^2+5x-12$$Now if I was given the last trinomial, I think I would have a hard time writing it as (2x-3)(x+4). On the other hand, -12 = -3*4, so it would still involve looking for two numbers with product -12. Now what about the 5? Maybe a little trial and error would give -3? I don't know.

One thing is sure, though: there is only one way to do it! (FTA dictates it)