This week in Math 10, I have chosen to talk about hard-to-factor polynomials, or ‘ugly’ polynomials.

When factoring, there is a checklist of three things for easy solving of a polynomial expression:

• 1 thing in common (If all the numbers have a common factor, you can divide them all by that number to ease the next steps)
• 2 terms (If the polynomial is a binomial that consists of a variable and a negative number, it is possible that is a square of the factorization)
• 3 terms (If the polynomial is a trinomial written as where A and B are integers, it is possible that B is the product of numbers that can add to A, which would be the constants of the factorization)

Some polynomials cannot be solved by any of the above means however, such as…

This polynomial was formed using the FOIL method, which stands for…

Front
Outside
Inside
Last

Here it is illustrated on a factorized polynomial, with each line corresponding to the step of the same colour.

With each step, the numbers multiply together to ‘expand,’ then it can be simplified. For example, with the image shown to the left, it would expand to which simplifies via like terms (numbers with identical exponents and/or variables added together) to

Back to , the first method we can use is to perform inspection, which is to say, brute force multiply the combinations of the factors the first and third terms with a FOIL problem until numbers whose sum is equal to the second term appear, or all combinations are exhausted. If the latter occurs, the number is not factorable.

The factors of the terms are…

: 3x, x
3: 3, 1

Now, we plug them into the FOIL problem to inspect. As the second term is negative, the factors of the third term will also be negative. We will try different combinations until numbers whose sum is -10 is found.

-3x-3x=-6x
This obviously wasn’t the right combination, so we’ll try another.

-x-9x=-10x
We now have the correct combination, which means the factorization is

The other method is to use an area model, which looks like this with the factorization we found.

Normally, we would put the multiplications of the numbers in each cell based on which column and row intersect with the cell. However, when factoring, the table would look more like the image below.

The bottom left and top right numbers are numbers can sum up the second term, based on whether they share factors with the first and third terms. The fully filled table would look like this: