We want to factor 18𝑥 to the fourth power minus 48𝑥 squared plus 30𝑥 into two factors. Given that one of these factors is three 𝑥 squared plus three 𝑥 minus five, what is the other factor?
So in this question, we’re told that there are two factors. One of them is three 𝑥 squared plus three 𝑥 minus five and the other one we need to work out. The answer to these two factors multiplied will be 18𝑥 to the fourth power minus 48𝑥 squared plus 30𝑥. Let’s imagine we have a simpler question. If we were told that we have two factors and one of them is 22, and they’re multiplied to give us 264. We could find out the other factor in a very quick way. We would simply calculate 264 divided by 22. So in this question, we’re going to take our polynomial, 18𝑥 to the fourth power minus 48𝑥 squared plus 30𝑥, and divide it by three 𝑥 squared plus three 𝑥 minus five. And, we can do this by long division.
So, we set out our long division by writing our dividend underneath and inside our dividing lines, and writing our divisor on the right side. It can be helpful to leave a gap where we’re missing a third power of 𝑥, which will be useful whenever we come to do our calculations. So, when it comes to long division of polynomials, we’re primarily concerned with the highest powers. So, 18𝑥 to the power of four in our dividend and three 𝑥 squared in our divisor.
So, the first thing we do is divide 18𝑥 to the fourth power by three 𝑥 squared. Or alternatively, we can think about it as three 𝑥 squared multiplied by what will give us 18𝑥 to the fourth power. So, since three times six gives us 18, the coefficient will be six. And we must have 𝑥 squared, since when we multiply 𝑥 squared by 𝑥 squared, we add the exponents two and two to give four. Which means that we have 𝑥 to the power of four. So, the answer to 18𝑥 to the fourth power divided by three 𝑥 squared is six 𝑥 squared.
Next we take our answer, six 𝑥 squared, and we multiply it by every term in the divisor, beginning with three 𝑥 squared. Since we’ve already worked out three 𝑥 squared times six 𝑥 squared gives us 18𝑥 to the fourth power, we write that in below. Next, we multiply six 𝑥 squared by three 𝑥, which gives us 18𝑥 to the power of three. And, we can write our answer in below the gap that we created when we set our calculation. And then, our final term in the divisor will multiply six 𝑥 squared by negative five, which gives us negative 30𝑥 squared.
So in our next stage, we need to subtract our values here from our dividend. To begin, we can subtract our 18𝑥 to the fourth power from 18𝑥 to the fourth power, which gives us zero. Since we had no third power in our dividend, we have zero take away 18𝑥 to the third power, which leaves us with negative 18𝑥 to the third power. In the next column, we have negative 48𝑥 squared take away negative 30𝑥 squared, which the same as negative 48𝑥 squared plus 30𝑥 squared. So, our answer to that is a negative 18𝑥 squared. And finally, since we have no term in 𝑥, this will leave us with 30𝑥 take away zero, which is plus 30𝑥.
And so, we repeat the process again and we’re checking our higher powers. This time, we’re saying, what value do we multiply three 𝑥 squared by to get negative 18𝑥 to the third power? Our answer to this must be negative six 𝑥, since three times negative six gives us negative 18, and 𝑥 squared times 𝑥 will give us 𝑥 to the third power. And now, we take our value, negative six 𝑥, and we multiply it by every term in the divisor, beginning with three 𝑥 squared. So, our first value will be negative 18𝑥 to the third power. Then, we multiply negative six 𝑥 by three 𝑥 giving us negative 18𝑥 squared. And finally, negative six 𝑥 times negative five will give us plus 30𝑥.
And in our final step, we subtract the bottom line from the line above it. As we have negative 18𝑥 to the third power take away negative 18𝑥 to the third power, we know that this cancels, giving us zero. Again, we have negative 18𝑥 squared take away negative 18𝑥 squared, which cancels, giving us zero. And finally, 30𝑥 take away 30𝑥 is also zero, which means that we’re left with no remainder. So, we can read off our final answer from our division calculation. Therefore, the missing factor is six 𝑥 squared minus six 𝑥.