Video: Dividing Third-Degree Polynomials Using Long Division to Find a Factor of a Polynomial

We want to factor 18π‘₯⁴ βˆ’ 48π‘₯Β² + 30π‘₯ into two factors. Given that one of these factors is 3π‘₯Β² + 3π‘₯ βˆ’ 5, what is the other?

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Video Transcript

We want to factor 18π‘₯ to the fourth power minus 48π‘₯ squared plus 30π‘₯ into two factors. Given that one of these factors is three π‘₯ squared plus three π‘₯ minus five, what is the other factor?

So in this question, we’re told that there are two factors. One of them is three π‘₯ squared plus three π‘₯ minus five and the other one we need to work out. The answer to these two factors multiplied will be 18π‘₯ to the fourth power minus 48π‘₯ squared plus 30π‘₯. 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, 18π‘₯ to the fourth power minus 48π‘₯ squared plus 30π‘₯, and divide it by three π‘₯ squared plus three π‘₯ 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 π‘₯, 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, 18π‘₯ to the power of four in our dividend and three π‘₯ squared in our divisor.

So, the first thing we do is divide 18π‘₯ to the fourth power by three π‘₯ squared. Or alternatively, we can think about it as three π‘₯ squared multiplied by what will give us 18π‘₯ to the fourth power. So, since three times six gives us 18, the coefficient will be six. And we must have π‘₯ squared, since when we multiply π‘₯ squared by π‘₯ squared, we add the exponents two and two to give four. Which means that we have π‘₯ to the power of four. So, the answer to 18π‘₯ to the fourth power divided by three π‘₯ squared is six π‘₯ squared.

Next we take our answer, six π‘₯ squared, and we multiply it by every term in the divisor, beginning with three π‘₯ squared. Since we’ve already worked out three π‘₯ squared times six π‘₯ squared gives us 18π‘₯ to the fourth power, we write that in below. Next, we multiply six π‘₯ squared by three π‘₯, which gives us 18π‘₯ 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 π‘₯ squared by negative five, which gives us negative 30π‘₯ squared.

So in our next stage, we need to subtract our values here from our dividend. To begin, we can subtract our 18π‘₯ to the fourth power from 18π‘₯ to the fourth power, which gives us zero. Since we had no third power in our dividend, we have zero take away 18π‘₯ to the third power, which leaves us with negative 18π‘₯ to the third power. In the next column, we have negative 48π‘₯ squared take away negative 30π‘₯ squared, which the same as negative 48π‘₯ squared plus 30π‘₯ squared. So, our answer to that is a negative 18π‘₯ squared. And finally, since we have no term in π‘₯, this will leave us with 30π‘₯ take away zero, which is plus 30π‘₯.

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 π‘₯ squared by to get negative 18π‘₯ to the third power? Our answer to this must be negative six π‘₯, since three times negative six gives us negative 18, and π‘₯ squared times π‘₯ will give us π‘₯ to the third power. And now, we take our value, negative six π‘₯, and we multiply it by every term in the divisor, beginning with three π‘₯ squared. So, our first value will be negative 18π‘₯ to the third power. Then, we multiply negative six π‘₯ by three π‘₯ giving us negative 18π‘₯ squared. And finally, negative six π‘₯ times negative five will give us plus 30π‘₯.

And in our final step, we subtract the bottom line from the line above it. As we have negative 18π‘₯ to the third power take away negative 18π‘₯ to the third power, we know that this cancels, giving us zero. Again, we have negative 18π‘₯ squared take away negative 18π‘₯ squared, which cancels, giving us zero. And finally, 30π‘₯ take away 30π‘₯ 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 π‘₯ squared minus six π‘₯.

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