Choose which of these polynomials is a difference of squares?
X^2-12x-36
X^2+12x+36
X^2-36
X^2+36
Then factor the polynomial that is a difference of squares. Your answer should include a polynomial from the above options and its factored form.
The polynomial that is a difference of squares is:
X^2 - 36
This polynomial can be factored as (X + 6)(X - 6), since it is in the form a^2 - b^2 = (a + b)(a - b). The factors are (X + 6) and (X - 6).
The polynomial that is a difference of squares is:
X^2 - 36
To factor this polynomial, we can use the identity a^2 - b^2 = (a + b)(a - b).
In this case, a is X and b is 6.
So, the factored form of the polynomial X^2 - 36 is:
(X + 6)(X - 6)
To determine which polynomial is a difference of squares, we need to look for a polynomial of the form \(a^2 - b^2\). The difference of squares can be factored as \((a + b)(a - b)\).
Let's analyze each polynomial from the options provided:
1. \(x^2 - 12x - 36\) does not fit the form \(a^2 - b^2\) because it does not have two terms squared.
2. \(x^2 + 12x + 36\) does fit the form \(a^2 - b^2\) because it can be rewritten as \((x + 6)^2\).
3. \(x^2 - 36\) does not fit the form \(a^2 - b^2\) because it does not have two terms squared.
4. \(x^2 + 36\) does not fit the form \(a^2 - b^2\) because it does not have two terms squared.
Therefore, the polynomial that is a difference of squares is \(x^2 + 12x + 36\).
Now, let's factor \(x^2 + 12x + 36\) using the difference of squares formula. We have \(a = x\) and \(b = 6\). Applying the formula, we have:
\((x + 6)(x + 6)\)
Simplifying this, we obtain:
\((x + 6)^2\)
So, the factored form of the polynomial \(x^2 + 12x + 36\) is \((x + 6)^2\).