Simplifying a rational expression, and the factor you must not cancel

0:00Here is a fraction with letters in it, and the instruction is to simplify. x squared minus seven x plus twelve, over five x squared plus five x minus a hundred. Which sounds like tidying up. It is not. Simplifying means finding something that is a factor of the top and a factor of the bottom, and dividing both by it. Dividing. Hold on to that word, because it is going to matter more than anything else here.

0:32There is exactly one number you are never allowed to divide by. And an expression with a letter in it can quietly become that number. So before any of this is allowed, something has to promise that the bottom is not nought. Take the top first. x squared minus seven x plus twelve. You want two numbers that add to minus seven and multiply to twelve. Search every whole number pair, and exactly one comes back. Minus four and minus three.

1:07So the top is x minus three, times x minus four. Multiply that out again and it lands on what you started with, term for term. That check costs nothing. Do it every time. Now the bottom. Five x squared plus five x minus a hundred. Do not go straight at the middle term. Five, five and a hundred are all multiples of five. Lift the five out and what is left inside is x squared plus x minus twenty, which is the easy kind.

1:43Two numbers adding to one and multiplying to minus twenty. Minus four and five. So the bottom is five, times x minus four, times x plus five. Taking the five out first was cheaper, and cheaper is not the same as necessary. Go at the middle term directly and you split minus five hundred into minus twenty and twenty five, and you land on exactly the same bottom. More work, same answer.

2:15Put them side by side and the shared bracket is impossible to miss. Compare the two lists of brackets and they have exactly one thing in common. That one. So cross it off, top and bottom, and what is left is x minus three over five, times x plus five. Except stop. You just divided by x minus four. How do you know that was not nought? Go back to the promise made at the very start, before a single bracket was drawn.

2:50The bottom is not nought. That sentence looks like throat clearing. It is the licence for everything that follows. Without it the cancelling is not a shortcut, it is an error. And notice where it sits: before the working, not after. Cancel first and look for permission later, and you have already done the thing you were not allowed to do. But the promise is about the whole bottom. The cancelling needs one bracket.

3:20So walk it across: the bottom is a product of five, x minus four and x plus five. If a product is not nought, then not one of the things multiplied is nought either. Check that everywhere in range and only two cases ever turn up. Either no bracket vanishes and the bottom is fine, or exactly one vanishes and the bottom is nought. There is never a third case.

3:49Now run the arrow backwards, and watch it fail. At x equals minus five, the bracket x minus four is perfectly healthy. It is minus nine. And the bottom is nought anyway, because the other bracket did it. One factor behaving tells you nothing about the product. The product behaving tells you about every factor. Here is the part nobody says out loud. The answer is not the same object as the question.

4:22The original has no value at four, and no value at minus five. Both of those kill its bottom. The simplified version still refuses minus five. But at x equals four it is perfectly happy, and it equals one forty-fifth. Run both across seventeen whole numbers and count. They agree at fifteen. At one value neither has anything to say. At exactly one, the second has an answer and the first does not.

4:55So they are not equal. They agree everywhere the first makes sense, and the second accepts one extra value. That gap is precisely what the promise at the start was protecting. Now the thing that goes wrong most. Cancelling something that is not a factor. Look again: x squared is sitting on top and on the bottom, begging to be struck out. Strike them and you get minus seven x plus twelve, over five x minus a hundred.

5:29Test it at x equals one. The real thing is minus one fifteenth. The struck one is minus one nineteenth. Not close. Just different. Across the range the struck version is right once and wrong fourteen times. The one time it is right is at x equals nought, where the x squared it removed contributed nothing anyway. A coincidence is not a defence. You may cancel factors. A term is not a factor.

6:02Here is a trap that costs marks quietly. Thirty six s squared minus twelve s t plus t squared, over t squared plus two t s minus forty eight s squared. The top is six s minus t, all squared. The bottom, read as a quadratic in t, needs two numbers adding to two and multiplying to minus forty eight. Eight and minus six. So the bottom is t plus eight s, times t minus six s.

6:36And now look. Upstairs it is six s minus t. Downstairs it is t minus six s. Those are not the same bracket. One is the other one's negative. But the top is squared, and squaring throws the overall sign away, so those two squares are the same expression. Rewrite it that way and one copy cancels, leaving t minus six s over t plus eight s. On a grid wide enough to see it, the original refuses seven pairs and the simplified one refuses three. Four values got freed, and nothing else changed.

7:18Try this one. Four quadratics, two on top and two underneath. Factorise all four and something startling happens. The same four brackets appear above and below. x plus three, x minus two, x minus three, x minus four. Upstairs and downstairs, the identical product. Subtract one from the other and there is nothing left at all. So it simplifies to one. And yet it is not the number one. It has no value at minus three, at two, at three, and at four.

7:55At every single one of those four the top is nought as well. Four cases of nought over nought, and the same brackets caused both. There is no second kind of failure to sort out. Thirteen values where it is one, four where it is nothing. An expression that simplifies to one is not equal to one. Do not learn the wrong lesson from the first example. p to the fourth minus sixteen, over p squared minus four p plus four.

8:28The top is a difference of squares twice over. p squared minus four, times p squared plus four. Then p minus two, times p plus two, times p squared plus four. The bottom is p minus two, squared. So one p minus two cancels. And one p minus two is still standing underneath. So p equals two was forbidden before, and it is forbidden afterwards. Nothing was freed. Put the two side by side. The first example gained a value. This one gained none.

9:07The rule is not that cancelling frees the value. It is that the original's forbidden values stay forbidden for the original, whatever the tidier version can do. One more, and it is the reason this comes at the end. w cubed minus v cubed plus x cubed plus three w v x, over the six-term expression underneath. Read the top with minus v in the middle slot and it is a cubed plus b cubed plus c cubed minus three a b c.

9:43So the top is w minus v plus x, times that long second bracket. And the bottom is the very same w minus v plus x, squared. One copy goes. You needed the cubic identity and the three-term square in the same breath, which is why nothing here could have come sooner. Last thing, because it is easy to assume otherwise. Sometimes there is no shared factor and the thing simply does not simplify.

10:14Of six worked quotients here, five share a bracket and one shares nothing at all. The condition is not only that the factor is not nought. It is also that there is one.