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Find the quotient when 16𝑥 power four plus 12𝑥 cubed minus 14𝑥 squared plus six 𝑥 is divided by four 𝑥 squared plus six 𝑥.
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To solve this problem, we’re actually gonna use long division.
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So I’m gonna to set that up now.
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Okay, so it’s now set up.
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We can actually get started.
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Well to find the first term of the quotient, what we actually need to do is we need to actually see what would multiply four 𝑥 squared by to get to our first term in our expression, which is 16𝑥 to the power of four.
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Well actually if we look at it, we’d actually see that we have to multiply four 𝑥 squared by itself, so four 𝑥 squared multiplied by four 𝑥 squared, because four multiplied by four gives us 16 and 𝑥 squared multiplied by 𝑥 squared gives us 𝑥 to the power four.
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So that means our first term of the quotient is gonna be four 𝑥 squared.
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Our next stage is actually to multiply this first term of our quotient by both the four 𝑥 squared and the six 𝑥.
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So we’ll start with the four 𝑥 squared multiplied by four 𝑥 squared, which gives us 16𝑥 power four.
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Then we move on, and we do four 𝑥 squared multiplied by six 𝑥.
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And this gives us plus 24𝑥 cubed, which I write underneath the 12𝑥 cubed.
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As you can see here what I’m actually doing is I’m keeping the powers of 𝑥 all in a line.
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So I’m keeping it nice and ordered.
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Okay, our next step is now to subtract the terms.
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Well first of all, we’d have 16𝑥 power four minus 16𝑥 power four.
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This actually give us an answer of zero.
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But you don’t necessarily have to write this.
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I’ve just put it there just to kinda highlight it in this first instance.
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And next, we’re gonna do 12𝑥 cubed minus 24𝑥 cubed.
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So this is gonna give us negative 12𝑥 cubed.
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Make sure we’re very careful with positives and negatives at this point.
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Excellent!
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So that’s the first stage completed.
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Now to move on to the next stage, what we do is actually bring down our next term, which in this case will be the negative 14𝑥 squared, and then we repeat the process again.
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So we start off by saying what do we need to multiply four 𝑥 squared by to get to negative 12𝑥 cubed.
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We can actually see we get negative three 𝑥 because negative three multiplied by four is negative 12 and 𝑥 squared multiplied by 𝑥 is 𝑥 cubed.
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So this is our next term in our quotient, again paying particular attention to our negative sign.
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And now we’re gonna multiply this second term by our terms of four 𝑥 squared and six 𝑥.
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So the first one we get is negative 12𝑥 cubed.
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That’s because it’s negative three 𝑥 multiplied by four 𝑥 squared.
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It’s worth noting at this point if for any reason the first bits aren’t the same — so you didn’t get 16𝑥 to the power four in both or negative 12𝑥 to the power of three — make sure that you go back and check your working because they should always be the same when we multiply it by the term in our quotient.
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Next, we’re gonna multiply our six 𝑥 by our negative three 𝑥.
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And this gives us negative 18𝑥 squared.
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So then, again, we’re gonna subtract these.
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So negative 12𝑥 cubed minus negative 12𝑥 cubed just gives us zero.
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And then we’re gonna do the next two terms.
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So we’ve got negative 14𝑥 squared minus negative 18𝑥 squared.
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And be careful here, again, because we got minus and negative so it’s actually going to be adding.
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So it’s gonna be negative 14𝑥 squared add 18𝑥 squared, which gives us four 𝑥 squared.
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Then, like the previous step, we actually bring down our next term.
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So in this time, it’s the 𝑥 term.
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So now we’ve got four 𝑥 squared plus six 𝑥.
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And now we’re gonna repeat that for a final time.
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Well here, we can say that it’s four 𝑥 squared multiplied by what gives us four 𝑥 squared.
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Well, we know it’s gonna be positive one because four 𝑥 squared multiplied by one is equal to four 𝑥 squared.
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So therefore, we can say the third term in our quotient is gonna be positive one.
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Then finally, we’re gonna multiply our terms.
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So we’ve got positive one multiplied by four 𝑥 squared, which therefore will give us four 𝑥 squared.
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Like I said, it’s the same as the previous one, which is what we wanted.
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Then we’re gonna do six 𝑥 multiplied by one, which gives us six 𝑥.
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So that means we can actually do our final round of subtractions.
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We got four 𝑥 squared minus four 𝑥 squared, which is zero, and six 𝑥 minus six 𝑥, which is zero.
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So therefore we can say there’s no remainder and that actually four 𝑥 squared plus six 𝑥 is a factor.
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And so therefore, we can say that the quotient, which is 𝑞𝑥, is equal to four 𝑥 squared minus three 𝑥 plus one.