# Question Video: Finding the Value That Makes a Polynomial Divisible by a Given Binomial Mathematics • 10th Grade

Find the value of π that makes the expression 30π₯β΅ + 57π₯Β² β 48π₯Β³ β 20π₯β΄ + π divisible by 5π₯Β² β 8.

04:58

### 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, letβ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, weβll make the problem a little bit easier, but it is going to look a little bit strange.

Letβ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 weβ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 weβre going to line that up directly underneath the π₯ cubed terms. And weβre going to add in another term; weβre going to add in zero π₯ to the fourth power. This isnβ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 weβ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, weβ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 weβre nearly there. We bring down the final term; thatβs the π.

Now, donβt worry too much that we donβt yet know what π is. Remember, weβre looking for a remainder of zero. Weβ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 weβre subtracting the final two terms, weβ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 letβ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. Thatβ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.