Completely factor 𝑥 cubed minus two 𝑥 squared plus five 𝑥 minus 10.
Here we can factor by grouping. So first we need to group the first two terms together and the last two terms together. But beforehand, we need to make sure that it’s in descending order. So the highest power needs to be first, and then next highest power second, and so on. And we have that with 𝑥 cubed, and then we have an 𝑥 squared, and then we have an 𝑥 to the first, even though there’s nothing there, that’s what represents a one, and then there’s no 𝑥, which is technically 𝑥 to the zero power because 𝑥 to zero power is one.
So as we said, we need to group the first two terms together and the last two terms together, and then take out a greatest common factor from each of them. So the GCF between these two terms would be 𝑥 squared. And if we take 𝑥 squared out of 𝑥 cubed, we would have 𝑥 left. So essentially, we’re taking 𝑥 cubed and dividing it by 𝑥 squared. And then if we would take negative two 𝑥 squared and take out an 𝑥 squared, we would have minus two.
So now we need to look at five 𝑥 minus 10. The GCF we can take out of this would be five. And if we take five out of five 𝑥, we would have 𝑥. And if we would take five out of negative 10, we would have minus two. So this is equal to what we’ve started with. And it’s very important that what is inside the parentheses match. They are both 𝑥 minus two, because these will become one of the factors. And putting the GCFs together will make a factor.
So 𝑥 squared and a positive five, we put together to make a factor and then the 𝑥 minus twos are identical so they make a factor. So 𝑥 squared plus five times 𝑥 minus two would be our final answer.