Question Video: Finding the Value That Makes a Polynomial Divisible by a Given Binomial Mathematics

Find the value of π‘˜ that makes the expression 30π‘₯⁡ + 57π‘₯Β² βˆ’ 48π‘₯Β³ βˆ’ 20π‘₯⁴ + π‘˜ divisible by 5π‘₯Β² βˆ’ 8.

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Video Transcript

Find the value of π‘˜ that makes the expression 30π‘₯ to the fifth power plus 57π‘₯ squared minus 48π‘₯ cubed minus 20π‘₯ to the fourth power plus π‘˜ divisible by five π‘₯ squared minus eight.

Now, for this expression to be divisible by five π‘₯ squared minus eight, that tells us that five π‘₯ squared minus eight is a factor of it. If this is the case, then when we perform the division, we should have no remainder, or a remainder of zero. So, let’s perform the long division. Now, if we look carefully at our expression, we see that the terms appear to be in a bit of a funny order. They are usually written in decreasing powers of π‘₯. And so, we might be tempted to rearrange them and write it as 30π‘₯ to the fifth power minus 20π‘₯ to the fourth power, and so on. Doing it this way, we’ll make the problem a little bit easier, but it is going to look a little bit strange.

Let’s begin as normal. We divide the term in our dividend with highest power of π‘₯ by the term in the divisor, also with the highest power of π‘₯. 30 divided by five is six. Then, when we’re dividing numbers whose bases are the same, we subtract their exponents. So, π‘₯ to the fifth power divided by π‘₯ squared is π‘₯ to the power of five minus two, which is π‘₯ cubed. This means that 30π‘₯ to the fifth power divided by five π‘₯ squared is six π‘₯ cubed. And so, we write six π‘₯ cubed above this term. We now multiply this value by each term in our divisor. Six π‘₯ cubed times five π‘₯ squared gives us 30π‘₯ to the fifth power. And then, when we multiply six π‘₯ cubed by negative eight, we get negative 48π‘₯ cubed.

Now we’re going to line that up directly underneath the π‘₯ cubed terms. And we’re going to add in another term; we’re going to add in zero π‘₯ to the fourth power. This isn’t entirely necessary, but it can make it a little bit easier to follow what happens next. Our next step is to divide each of these three terms by the corresponding terms above. 30π‘₯ to the fifth power minus 30π‘₯ to the fifth power is zero. Negative 20π‘₯ to the fourth power minus zero π‘₯ to the fourth power is negative 20π‘₯ to the fourth power. And negative 48π‘₯ cubed minus negative 48π‘₯ cubed is zero.

Next, we bring down 57π‘₯ squared. And we’re now going to divide negative 20π‘₯ to the fourth power by five π‘₯ squared. Negative 20 divided by five is negative four, and π‘₯ to the fourth power divided by π‘₯ squared is just π‘₯ squared. And so, when we do this division, we get negative four π‘₯ squared. And this is the next term in our quotient. Remember, each time we find a term in our quotient, we multiply it by each part of the divisor. So, we’re going to work out negative four π‘₯ squared times five π‘₯ squared, which is negative 20π‘₯ to the fourth power.

When we multiply negative eight by negative four π‘₯ squared, we get 32π‘₯ squared. So, we can line this up directly under 57π‘₯ squared. And then, we subtract each of these terms from the terms immediately above them. Negative 20π‘₯ to the fourth power minus negative 20π‘₯ to the fourth power is zero. Then, 57π‘₯ squared minus 32π‘₯ squared is 25π‘₯ squared. And we’re nearly there. We bring down the final term; that’s the π‘˜.

Now, don’t worry too much that we don’t yet know what π‘˜ is. Remember, we’re looking for a remainder of zero. We’re going to divide 25π‘₯ squared by five squared. Well, 25 divided by five is five, and π‘₯ squared divided by π‘₯ squared is one. So, the final term in our quotient is five. We take that five and we multiply it by five π‘₯ squared and negative eight. And that gives us 25π‘₯ squared minus 40.

Now, we know that since we’re subtracting the final two terms, we’re going to get a remainder of zero. So, 25π‘₯ squared plus π‘˜ minus 25π‘₯ squared minus 40 has to give us zero. Well, 25π‘₯ squared minus 25π‘₯ squared is also zero. And so to ensure that our remainder is zero, and thus our expression is divisible by five π‘₯ squared minus eight, we can say that π‘˜ minus negative 40 itself must be equal to zero. π‘˜ minus negative 40 is, of course, π‘˜ plus 40.

And so let’s subtract 40 from both sides to solve for π‘˜. That gives us π‘˜ is equal to negative 40. And so, the value of π‘˜ that makes our expression divisible by five π‘₯ squared minus eight is π‘˜ equals negative 40. Note that at this point, we could go ahead and multiply our quotient. That’s six π‘₯ cubed minus four π‘₯ squared plus five by five π‘₯ squared minus eight. If we had performed the calculation correctly, we would end up with the expression 30π‘₯ to the fifth power plus 57π‘₯ squared minus 48π‘₯ cubed minus 20π‘₯ to the fourth power minus 40.

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