The expression you’re working with is x² – 25 – 10x. To determine whether this is a difference of two squares, we first need to rearrange and simplify the expression.
We can rewrite the expression as:
x² – 10x – 25
Next, let’s look for a way to express this polynomial in the form of a difference of two squares, which is generally structured as a² – b² = (a + b)(a – b). To do this, we can complete the square for the quadratic part of the expression (x² – 10x).
Completing the square involves taking half of the coefficient of the x term (which is -10), squaring it, and then using that to rewrite the expression:
1. Half of -10 is -5, and squaring it gives us 25.
2. Now, we can rewrite our polynomial:
x² – 10x + 25 – 25
=> (x – 5)² – 25
This shows us that we can express our polynomial in the difference of squares form:
(x – 5)² – 5²
Now we can factor it using the difference of squares formula:
(x – 5 + 5)(x – 5 – 5)
=> (x)(x – 10)
In conclusion, the polynomial x² – 25 – 10x can indeed be seen as a difference of squares, which can be factored into (x)(x – 10).