factor the polynomials
x^2 -6
x^2 +5x-6
a^2 - b^2 = (a+b)(a-b)
x^2 - 6 = (x+sqrt 6)(x- sqrt 6)
x^2 + 5x - 6 = (x+6)(x-1)
To factor the given polynomials, we need to identify the factors that can multiply together to give us the original expression.
Let's start with the first polynomial: x^2 - 6.
To factor this polynomial, we're looking for two numbers whose product is -6 and sum is 0 (since there is no coefficient of x present). The factors of -6 that satisfy these conditions are -2 and 3.
Therefore, we can factor x^2 - 6 as (x - 2)(x + 3).
Now, let's move on to the second polynomial: x^2 + 5x - 6.
To factor this polynomial, we're looking for two numbers whose product is -6 and sum is 5 (the coefficient of x). The factors of -6 that satisfy these conditions are -6 and 1.
Now, we rearrange the polynomial by splitting 5x into the terms involving -6x and 1x:
x^2 - 6x + 1x - 6.
Next, we group the terms:
(x^2 - 6x) + (1x - 6).
Now we factor by taking out the common factors from each group:
x(x - 6) + 1(x - 6).
Notice that both groups have a common factor (x - 6), which we can then factor out:
(x - 6)(x + 1).
Therefore, we can factor x^2 + 5x - 6 as (x - 6)(x + 1).
In summary, the factored forms of the given polynomials are:
x^2 - 6 = (x - 2)(x + 3)
x^2 + 5x - 6 = (x - 6)(x + 1)