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

Video Transcript

Find the value of 𝑘 that makes the expression 30𝑥 to the fifth power plus 57𝑥 squared minus 48𝑥 cubed minus 20𝑥 to the fourth power plus 𝑘 divisible by five 𝑥 squared minus eight.

Now, for this expression to be divisible by five 𝑥 squared minus eight, that tells us that five 𝑥 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 𝑥. And so, we might be tempted to rearrange them and write it as 30𝑥 to the fifth power minus 20𝑥 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 𝑥 by the term in the divisor, also with the highest power of 𝑥. 30 divided by five is six. Then, when we’re dividing numbers whose bases are the same, we subtract their exponents. So, 𝑥 to the fifth power divided by 𝑥 squared is 𝑥 to the power of five minus two, which is 𝑥 cubed. This means that 30𝑥 to the fifth power divided by five 𝑥 squared is six 𝑥 cubed. And so, we write six 𝑥 cubed above this term. We now multiply this value by each term in our divisor. Six 𝑥 cubed times five 𝑥 squared gives us 30𝑥 to the fifth power. And then, when we multiply six 𝑥 cubed by negative eight, we get negative 48𝑥 cubed.

Now we’re going to line that up directly underneath the 𝑥 cubed terms. And we’re going to add in another term; we’re going to add in zero 𝑥 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. 30𝑥 to the fifth power minus 30𝑥 to the fifth power is zero. Negative 20𝑥 to the fourth power minus zero 𝑥 to the fourth power is negative 20𝑥 to the fourth power. And negative 48𝑥 cubed minus negative 48𝑥 cubed is zero.

Next, we bring down 57𝑥 squared. And we’re now going to divide negative 20𝑥 to the fourth power by five 𝑥 squared. Negative 20 divided by five is negative four, and 𝑥 to the fourth power divided by 𝑥 squared is just 𝑥 squared. And so, when we do this division, we get negative four 𝑥 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 𝑥 squared times five 𝑥 squared, which is negative 20𝑥 to the fourth power.

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

Now, don’t worry too much that we don’t yet know what 𝑘 is. Remember, we’re looking for a remainder of zero. We’re going to divide 25𝑥 squared by five squared. Well, 25 divided by five is five, and 𝑥 squared divided by 𝑥 squared is one. So, the final term in our quotient is five. We take that five and we multiply it by five 𝑥 squared and negative eight. And that gives us 25𝑥 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, 25𝑥 squared plus 𝑘 minus 25𝑥 squared minus 40 has to give us zero. Well, 25𝑥 squared minus 25𝑥 squared is also zero. And so to ensure that our remainder is zero, and thus our expression is divisible by five 𝑥 squared minus eight, we can say that 𝑘 minus negative 40 itself must be equal to zero. 𝑘 minus negative 40 is, of course, 𝑘 plus 40.

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