Let’s factor 𝑥 squared minus 𝑥 minus 30.
Immediately, we notice that our 𝑎-value, or our leading coefficient is one. This makes the process
of factoring a lot easier. We actually don’t need to do anything with our leading coefficient now,
so we can make that go away. Since our leading coefficient is one, we can go ahead and say 𝑥
times 𝑥 will be the first term in each of these factors. Finding the second term in these factors will
be a little bit trickier. What we need for the second terms will be factors of 30 that add up to
negative one. Our factors will need to multiply together to equal negative 30 and add together
to be equal negative one.
We can start this process by making a list of the factors of 30. One and 30, two and 15, three and
10. 30 is not divisible by four, but five and six are also factors of 30. Now I wanna look carefully
at these factors to see if any of them can add up to equal negative one. One plus 30 wouldn’t
work. One plus negative 30 would not work. Negative one plus 30 also would not work. One and
30 isn’t the right option for us. And because one and 30 didn’t work, I’m pretty sure that two and
15 won’t work either because their numbers are too far apart. We need numbers that are closer
Let’s see if we can make five and six work. So five plus six equals 11; that doesn’t work. Negative five
plus six equals one, so we’re getting closer. Five plus negative six equals negative one. These will be
our second terms. We would be working with a positive five and a negative six. So we found the two
factors, 𝑥 plus five times 𝑥 minus six.
If we wanted to check and make sure that we’ve done everything correctly, we could foil this out to
see if it was true. We would start by multiplying 𝑥 times 𝑥, which equals 𝑥 squared. 𝑥 times six, sorry 𝑥
times negative six would equal negative six 𝑥. Five times 𝑥 equals five 𝑥, and five times negative six
equals negative 30. When we simplify that, we can say negative six 𝑥 plus five 𝑥 equals negative 𝑥.
And we could bring down our negative 30.
So after we multiply it out, we found what we originally started with which means that we have
correctly factored 𝑥 squared minus 𝑥 minus 30. The factorization would be 𝑥 plus five times
𝑥 minus six.