x² + 10x + 25 – 9y² I am asked to state if this polynomial is prime or to factor it. I can clearly see that you can take the SQRT of the equation, however, there is a 10x there that is not a perfect square. What am I doing wrong?
(x + 5)^2 - (3 y)^2 ... factor the difference of squares
In order to determine if the given polynomial is prime or if it can be factored, it is necessary to analyze it further. I notice that the polynomial you provided, x² + 10x + 25 – 9y², is not a perfect square trinomial. However, it can still be simplified and factored using other techniques.
To begin, we can separate the polynomial into two groups: the terms involving x (x² + 10x) and the terms not involving x (-9y² + 25).
Looking at the first group, x² + 10x, we can observe that it is indeed not a perfect square trinomial. It cannot be factored using simple square roots because there is an extra term, the 10x. Generally, perfect square trinomials have the form (x + a)² or (x - a)², where a is a constant. In this case, the middle term 10x is not of that form, so square roots cannot be helpful.
On the other hand, let's examine the second group, -9y² + 25. This is a simple quadratic binomial, and we can re-write it as 25 - 9y². Notice that it has the form a² - b², which can be factored into (a + b)(a - b) using the difference of squares formula. In this case, a = 5 and b = 3y, so we have:
25 - 9y² = (5 + 3y)(5 - 3y)
Now, let's put the two groups back together:
x² + 10x + (25 - 9y²) = x² + 10x + (5 + 3y)(5 - 3y)
Thus, the factored form of the given polynomial is x² + 10x + (5 + 3y)(5 - 3y). It is not a prime polynomial since it can be expressed as the product of two binomials.
To summarize the steps:
1. Separate the terms involving x and the terms not involving x.
2. Examine the term group involving x to determine if it is a perfect square trinomial (x² + 2ax + a²).
3. Apply the difference of squares formula to factor the term group not involving x (a² - b² = (a + b)(a - b)).
4. Reconstruct the factored form by combining the two groups.
Remember to always consider different factoring techniques, such as the difference of squares, when faced with polynomials that do not fit directly into basic factoring patterns.