I have a question I hope someone can explain it to me.
We are working on Demonstrate that f actoring a polynomial. I was ask Why can you factor x² - 4 but you cannot factor x² + 4? How can you tell quickly which ones you can factor and which you cannot? I am not sure what to answer here. I feel like it has some thing to do with the GCF but I am sure can some just give me an example and explain to me.
actually, one can factor x^2 + 4
(x+2i)(x-2i) are the factors, where i is the sqrt of -1.
The reality is that you cant look at a polynomial and know how many real factors there are. Experience helps, but when one has
x^3+3x^2-3x-14 there is no easy way to know without some detailed examination.
The X^2 - 4 can be factored because the negative sign allows us to make one factor + and the other one - as in
(x+2)(x-2) which when we expand it has the middle term cancel--to wit:x^2 +2x-2x-4.
When we try to factor x^2+4, we KNOW both factors must be EITHER + or - because + x + = + and - x - = +. As long as the two factors have the same sign, we always get a middle term BUT x^2+4 has no middle term.
So if I understanding this for example I have x^2-6 I can factor because the negative sign I can get a middle term
which would be
(x+2)(x-3) expand it to
(x^2 +2x-3x-6)
(x^(2)-x-6)
With x^2+6 I could not get a middle term because the sign is positive which shows me that because When you multiply two + togther you a + and when you mulitiply two - togther you get a + so this can not have a middle term
Did I understand this correct
To determine whether a quadratic polynomial can be factored, you need to check its discriminant. The discriminant of a quadratic polynomial of the form ax² + bx + c is given by the expression Δ = b² - 4ac.
1. In the case of x² - 4, the discriminant is Δ = 0² - 4(1)(-4) = 16. Since the discriminant is positive, we can factor the polynomial. To factorize x² - 4, we observe that it is in the form of a difference of squares, which can be factored as (x + 2)(x - 2).
2. Now, let's consider x² + 4. In this case, the discriminant is Δ = 0² - 4(1)(4) = -16. Since the discriminant is negative, we cannot factor the polynomial using real numbers. However, if you allow complex numbers, you can factorize it as (x + 2i)(x - 2i), where i is the imaginary unit (√-1).
Therefore, the key factor in determining whether a quadratic polynomial can be factored quickly is the discriminant. If the discriminant is positive, the polynomial can be factored. If the discriminant is zero, it is a perfect square and can be factored as a difference of squares. If the discriminant is negative, the polynomial cannot be factored using real numbers, but it can be factored using complex numbers.