Question Video: Using Polynomial Division to Solve Problems Mathematics

Find the quotient when π‘₯⁴ βˆ’ 8π‘₯Β² + 20π‘₯ βˆ’ 21 is divided by π‘₯Β² + 2π‘₯ βˆ’ 7.

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Video Transcript

Find the quotient when π‘₯ to the power of four minus eight π‘₯ squared plus 20π‘₯ minus 21 is divided by π‘₯ squared plus two π‘₯ minus seven.

In order to solve this problem, I’ve set up long division. And it’s gonna help us to find the quotient. One thing to note though that actually when I’ve set up the long division, I’ve included this term, which is zero π‘₯ cubed. So we actually include the zero π‘₯ cubed just so that everything keeps ordered and inline. If you have any missing terms, you can just insert the zero version, so like here zero π‘₯ cubed, just to help you with that.

The first step we can see here is we need to see what π‘₯ squared β€” so that’s the first term β€” in our divisor is multiplied by to give us π‘₯ to the power of four, which is the first term in our dividend. We’ll catch to work out that π‘₯ squared multiplied by π‘₯ squared will be equal to π‘₯ to the power of four. And that’s because π‘₯ squared multiplied by π‘₯ squared add the powers gives us π‘₯ to the power of four. So great! We’ve got the first term in our quotient.

The next step is to actually multiply our first term in our quotient by each of the terms in our divisor. So first of all, we have π‘₯ squared multiplied by π‘₯ squared, which gives us π‘₯ to the power of four. Then, we have positive two π‘₯ multiplied by π‘₯ squared, which gives us positive two π‘₯ cubed. And then finally, we have negative seven multiplied by π‘₯ squared, which gives us negative seven π‘₯ squared. And as you can see, each time I’ve actually worked these out, I’ve written them beneath the equivalent power of π‘₯ in the line above on our dividend.

The next stage is to actually subtract the terms that we’ve just created from the terms immediately above them. So first of all, we’d have π‘₯ to the power of four minus π‘₯ to the power of four, which would just be zero. Then, we have zero π‘₯ cubed minus two π‘₯ cubed, which gives us negative two π‘₯ cubed. So that’s why we put the zero in like I said to keep things aligned.

Next one we have is negative eight π‘₯ squared minus negative seven π‘₯ squared. So that’s gonna be the same as adding on seven π‘₯ squared to negative eight π‘₯ squared, which gives us negative π‘₯ squared. Okay, great! Our next step is actually to bring the next term from the dividend down, which in this case is going to be positive 20π‘₯.

Okay, fantastic! We can now start to repeat the steps again. So this time, we’re gonna be looking and see what π‘₯ squared, so the first term in our divisor, is gonna be multiplied by to give us negative two π‘₯ cubed. Well, we can see that it’s gonna be negative two π‘₯ multiplied by our π‘₯ squared to give us our negative two π‘₯ cubed because we’ve got the negative two and then π‘₯ multiplied by π‘₯ squared gives us π‘₯ cubed. So this term becomes the next term in our quotient.

Then, we repeat a step from earlier. And we actually multiply the negative two π‘₯, so the second term in our quotient, by the π‘₯ squared, which is the first term in our divisor, which will give us a negative two π‘₯ cubed. And it’s worth noting at this point that at each line of the working, we should have the same term at the start. So for instance, we had π‘₯ to the power of four minus π‘₯ to the power of four here. We’ve got negative two π‘₯ cubed minus negative two π‘₯ cubed. If it isn’t the case, then check your working, so there may be an error somewhere.

Next, we multiply the negative two π‘₯ in our quotient by positive two π‘₯ here in the divisor. This gives us negative four π‘₯ squared. And then, the final one which is negative two π‘₯ multiplied by negative seven, which gives us positive 14π‘₯.

Okay, great! So now, we’re gonna subtract. So we’ve got negative two π‘₯ cubed minus negative two π‘₯ cubed, which is just zero. Then, negative π‘₯ squared minus negative four π‘₯ squared, which gives us three π‘₯ squared. So again be careful of the negative here cause it’s minus a negative. Then, 20π‘₯ minus 14π‘₯, which just gives us our positive six π‘₯. Then, we bring down the final term in the dividend, which is negative 21. And we repeat the process for the final time. So we got π‘₯ squared multiplied by something gives us three π‘₯ squared. Well, it’s gonna be π‘₯ squared multiplied by three gives us three π‘₯ squared. So therefore, the final term in our quotient is positive three.

And then now, we’re gonna multiply this positive three by each term in our divisor. So starting with three multiplied by π‘₯ squared, this gives us three π‘₯ squared. Three multiplied by positive two π‘₯ gives us positive six π‘₯. Then finally, positive three multiplied by negative seven gives us negative 21.

So great! We can do our final subtraction. And we have three π‘₯ squared minus three π‘₯, which is just zero, six π‘₯ minus six π‘₯ again, which is zero, and negative 21 minus negative 21. We’re gonna get a total remainder of zero. So therefore, we can actually say that π‘₯ squared plus two π‘₯ minus seven is a factor.

So finally, we can say that the quotient when π‘₯ to the power of four minus eight π‘₯ squared plus 20π‘₯ minus 21 is divided by π‘₯ squared plus two π‘₯ minus seven is equal to π‘₯ squared minus two π‘₯ plus three.

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