We have to factor negative 35𝑥 cubed plus seven 𝑥 squared minus 42𝑥 to the four into two factors. Given that one of these factors is 𝑥 minus six 𝑥 squared, what is the other?
We have this polynomial. And we’re told that this has two factors, one of which is 𝑥 minus six 𝑥 squared. And the question is what is the other factor. Well, we can set this up as a polynomial long division problem.
If we divide this polynomial by one of its factors, then the quotient should be the other factor. But before we start this long division process, we need to rearrange the terms of both divisor and dividend. We would prefer the terms in a divisor to be written from highest degree to lowest degree.
So rather than writing 𝑥 minus six 𝑥 squared, we should write negative six 𝑥 squared plus 𝑥. This represents the same polynomial, but now the 𝑥 squared term come before the 𝑥 term. And the same is true of the dividend. We have the 𝑥 to the four term at the end when it should be in front of the 𝑥 cubed and 𝑥 squared terms.
Now the terms are written in the right order. And so we can start the division process. We divide the highest-degree term of the dividend, negative 42𝑥 to the four, by the highest degree term of the divisor, negative six 𝑥 squared.
It would have been harder to work out which terms we needed to divide had the terms of the divisor and dividend not been written in order. Dividing, we get seven 𝑥 squared which forms part of our quotient. And we put that on top. We now need to subtract seven 𝑥 squared times the divisor, negative six 𝑥 squared plus 𝑥, from the dividend.
But before we do that, we expand the parentheses. So we subtract negative 42𝑥 to the four plus seven 𝑥 cubed. And subtracting, we get negative 42𝑥 cubed plus seven 𝑥 squared.
Notice how the like terms in our subtraction problem lined up nicely. This is because we took the time to order the terms in the divisor and dividend. We’ve done one round, but we need to keep going. Our new dividend is negative 42𝑥 cubed plus seven 𝑥 squared.
The highest term of this is negative 42𝑥 cubed. And we divide this by the highest-degree term of the divisor, negative six 𝑥 squared. Performing this division, we get seven 𝑥 which we add to the quotient.
And now we need to subtract seven 𝑥 times our divisor. Expanding, we find that we need to subtract negative 42𝑥 cubed plus seven 𝑥 squared, which of course gives us zero.
Now we stop dividing and look at what we’ve got. We have a remainder of zero and a quotient of seven 𝑥 squared plus seven 𝑥. So our mystery factor is seven 𝑥 squared plus seven 𝑥.