To factor the polynomial x^2 - x - 6, we need to find two binomials whose product is equal to this polynomial. We can use the following methods to factor the polynomial:
Method 1: Factoring by inspection
- We know that x^2 is the product of x and x, so we can start with the binomial (x )(x ) as a factorization of x^2.
- We then look for two numbers whose product is -6 and whose sum is -1.
- The two numbers are -3 and 2, since (-3)(2) = -6 and (-3) + 2 = -1.
- Therefore, the polynomial x^2 - x - 6 can be factored as (x - 3)(x + 2).
Method 2: Using the quadratic formula
- We can also use the quadratic formula to find the roots of the polynomial, which are the values of x that make the polynomial equal to zero.
- The quadratic formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a, where a = 1, b = -1, and c = -6.
- Plugging in these values, we get x = (-(-1) ± sqrt((-1)^2 - 4(1)(-6))) / 2(1) = (1 ± sqrt(25)) / 2.
- Simplifying, we get x = 3 or x = -2.
- Therefore, the polynomial x^2 - x - 6 can be factored as (x - 3)(x + 2).
So, the pairs of polynomials that, when multiplied together, result in the polynomial x^2 - x - 6 are (x - 3) and (x + 2), as well as (x + 2) and (x - 3).