factor the polynomials
x^2 -6
x^2 +5x-6
To factor the polynomials, let's look at each one individually:
1. x^2 - 6:
First, check if it is a perfect square trinomial. In this case, it is not. So, we need to use the difference of squares formula.
The difference of squares formula is: a^2 - b^2 = (a + b)(a - b).
Now, let's apply the formula to x^2 - 6:
x^2 - 6 = (x + √6)(x - √6).
So, the factored form of x^2 - 6 is (x + √6)(x - √6).
2. x^2 + 5x - 6:
We need to find two numbers whose product is -6 and whose sum is 5 (the coefficient of the middle term).
After some trial and error, we find that 6 and -1 satisfy these conditions since 6 * -1 = -6 and 6 + (-1) = 5.
Now, we can rewrite the quadratic expression as follows:
x^2 + 5x - 6 = x^2 + 6x - x - 6.
Next, let's group the terms and factor by grouping:
(x^2 + 6x) - (x + 6) = x(x + 6) - 1(x + 6).
Now, factor out the common binomial:
(x + 6)(x - 1).
So, the factored form of x^2 + 5x - 6 is (x + 6)(x - 1).