How do you factor a polynomial?
Using the FOIL method works with some simple polynomials. FOIL stands for First, Outside, Inside, Last.
For example:
x^2-4x+4 factors to
(x-2)(x-2)
Check your work by multiplying them all. x times x is first, x times -2 is outside, -2 times x is inside, and -2 times -2 is last. Add the terms to get x^2-2x-2x+4 or x^2-4x+4
In the blank equation, (x+a)(x+b), we see that for this type of polynomial ax and bx have to add to make the middle term and a and b have to multiply to equal the final term. Just pick numbers which suit these conditions. It's easy with practice. Difficult to show.
Factoring a polynomial can be done using different methods, but I'll explain one common approach called factoring by grouping. Here are the steps:
1. Look for a common factor: Start by checking if there is a common factor among all the terms in the polynomial. If there is, factor it out.
2. Group the terms: Next, group the remaining terms in pairs. This is done by looking for pairs of terms that have a common factor.
3. Factor out the common factor: Within each group, factor out the common factor.
4. Look for similarities: Check if the resulting factors in each group share any similarities. If there is a common binomial factor, factor it out.
5. Combine the factored terms: Finally, combine the factored terms to form the fully factored polynomial.
Let's look at an example to illustrate the steps. Consider the polynomial: x^3 - 3x^2 + 2x - 6
1. In this case, there is no common factor to factor out.
2. Group the terms: Group the first two terms and the last two terms: (x^3 - 3x^2) + (2x - 6)
3. Factor out the common factor: In the first group, we can factor out x^2, and in the second group, we can factor out 2.
(x^2 (x - 3)) + (2(x - 3))
4. Look for similarities: Notice that (x - 3) is a common binomial factor that can be factored out.
(x^2 + 2) (x - 3)
5. Combine the factored terms: The fully factored polynomial is (x^2 + 2)(x - 3).
It's important to note that factoring polynomials can be more complex, especially for higher-degree polynomials or if there are no obvious factors. In such cases, other techniques like factoring by using the rational roots theorem or completing the square may be required.