How can you determine if a polynomial is the difference of two squares? I think that if the polynomial is x^2-64 that this is a polynomial that is a difference of two squares because if you factor it out you would get
(x+8)(x-8).
If the 64 in my example were positive would there still be a difference between two squares since x^2 and 64 both are squared numbers?
No, when it says difference, it needs the single - sign.
(a-b)(a+b) = a^2 + ab -ab - b^2 = a^2-b^2
but
(a+b)(a+b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 which is no good
Oh and x^2+64
solve x^2 + 0x + 64 = 0
x = [-0 +/- sqrt (0 -256) / 2
x = +/- sqrt (-256) /2
x = +/- 8 i
so
x*2+64 = (x-8i)(x+8i)
In other words there is no real number solution, You need the sqrt of -1 or "i"
To determine if a polynomial is the difference of two squares, you need to factor it completely. If the polynomial can be factored into the product of two binomials, each having a perfect square term and an opposite sign constant term, then it is a difference of two squares.
Let's use your example to understand this concept. You have the polynomial x^2 - 64. To check if it is a difference of two squares, we can try factoring it.
First, observe that 64 is a perfect square because it can be expressed as 8^2.
Now, let's factor the polynomial: x^2 - 64.
We can rewrite it as (x^2 - 8^2).
We recognize that this expression is in the form of a difference of squares: (a^2 - b^2) = (a + b)(a - b).
Using this formula, we can factor the expression as (x + 8)(x - 8).
So, x^2 - 64 can be expressed as (x + 8)(x - 8), which satisfies the criteria for a difference of two squares.
Now, let's address your question about whether there would still be a difference of two squares if the constant term (in this case, 64) were positive.
Yes, even if the constant term is positive, the polynomial can still be a difference of two squares. The sign of the constant term doesn't affect the nature of the polynomial.
For example, let's consider the polynomial x^2 - 49. We can factor it as (x + 7)(x - 7), which is again a difference of two squares, even though the constant term 49 is positive.
So, the key determining factor is the ability to factor the polynomial into two binomials with perfect square terms and opposite sign constant terms, regardless of whether the constant term is positive or negative.