If we divide the polynomial by the expression and there's no remainder, then we've found a factor.Īn easier way is to make use of the Remainder Theorem, which we met in the previous section, Factor and Remainder Theorems. We'd need to multiply them all out to see which combination actually did produce p( x).įinding one factor: We try out some of the possible simpler factors and see if the "work". However, it would take us far too long to try all the combinations so far considered. We observe the −6 as the constant term of our polynomial, so the numbers b, d, and g will most likely be chosen from the factors of −6, which are ☑, ☒, ☓ or ☖. This has to be the case so that we get 4 x 3 in our polynomial. The factors of 4 are 1, 2, and 4 (and possibly the negatives of those) and so a, c and f will be chosen from those numbers. We are looking for a solution along the lines of the following (there are 3 expressions in brackets because the highest power of our polynomial is 3):Ĥ x 3 − 3 x 2 − 25 x − 6 = ( ax − b)( cx − d)( fx − g) Notice the coefficient of x 3 is 4 and we'll need to allow for that in our solution. How to factor polynomials with 4 terms? Example 3Ībove, we discussed the cubic polynomial p( x) = 4 x 3 − 3 x 2 − 25 x − 6 which has degree 3 (since the highest power of x that appears is 3). Notice our 3-term polynomial has degree 2, and the number of factors is also 2. We say the factors of x 2 − 5 x + 6 are ( x − 2) and ( x − 3). The number 6 (the constant of the polynomial) has factors 1, 2, 3, and 6 (and the negative of each one is also possible) so it's very likely our a and b will be chosen from those numbers. This generally involves some guessing and checking to get the right combination of numbers. We need to find numbers a and b such that
We recognize this is a quadratic polynomial, (also called a trinomial because of the 3 terms) and we saw how to factor those earlier in Factoring Trinomials and Solving Quadratic Equations by Factoring. Here's an example of a polynomial with 3 terms: How to factor polynomials with 3 terms? Example 2 Continue, until we get to a trinomial, which we can usually factor easily.Find one factor of the simpler polynomial, and divide once again.Divide the polynomial by the factor we found, thus giving us a simpler polynomial to work with.Find one factor, by making use of the Remainder Theorem.We'll see how to find those factors below, in How to factor polynomials with 4 terms? Summary of the process When we multiply those 3 terms in brackets, we'll end up with the polynomial p( x). Note there are 3 factors for a degree 3 polynomial. What are we looking for? Example 1Īn example of a polynomial (with degree 3) is: We'll make use of the Remainder and Factor Theorems to decompose polynomials into their factors. On this page we learn how to factor polynomials with 3 terms (degree 2), 4 terms (degree 3) and 5 terms (degree 4).