Why a root of the equation is the same thing as a zero of the polynomial

0:00Two sentences, about the same number. One is a claim about an equation: this number is a ROOT of two x squared minus three x plus one equals nought. The other is a claim about an expression: this number is a ZERO of two x squared minus three x plus one. They look like two different findings that happen to agree. They are not. They are one condition written twice, and this video is about why.

0:33Start with a number and a question. Is one a root of two x squared minus three x plus one equals nought? There is a way to find out that needs no method at all. Put it in. Two times one squared is two. Minus three times one is minus three. And the last term is one. Two, minus three, one. They total nothing. So one is a root. That is the whole test.

1:01Now look at what you actually did. You never used the equals sign. You took the expression, replaced x by one, and found the value was nothing. Which is exactly what it means to say one is a ZERO of that expression. One substitution. Two names. And nothing was done twice. Before going on, it is worth watching the test fail, because every example you are likely to meet uses a number that works.

1:33A check that has never rejected anything is not a check. It is a ceremony. So try two in the same expression. Two times two squared is eight. Minus three times two is minus six. The last term is one. Eight, minus six, one. That totals three. Three is not nothing. So two is not a zero of the expression, and it is not a root of the equation either.

2:01Both verdicts, again, from one substitution. And notice how little it cost: three terms and two additions. Here is the general statement, and it uses a second letter on purpose. Let alpha be a real number, and take a quadratic equation in standard form: a x squared plus b x plus c equals nought. Alpha is a root of that equation exactly when a alpha squared plus b alpha plus c comes out as nothing.

2:36The letter changes because the two roles are different. x is the variable, the empty slot. Alpha is a particular number being tried in it. Keeping them apart on the page keeps them apart in your head. And read the statement again: it says nothing about solving. It describes a property a number either has or does not have. Three words get used for this, and they are worth sorting out, because they attach to different objects.

3:09A ROOT is a number. It belongs to the number. A SOLUTION is what that number is OF — it belongs to the equation. And SATISFIES is the verb: the number satisfies the equation. One condition, three grammatical homes. Nothing distinguishes them mathematically. A ZERO is the fourth word, and it belongs to the polynomial rather than to the equation. Which is the only reason there are two words for one thing at all.

3:42So why are the zeroes and the roots the same numbers? Not because someone proved it. Because of what the equation says. Setting the polynomial equal to nought is a request: find the numbers at which this expression comes out as nothing. And the numbers at which a polynomial comes out as nothing are called its zeroes. The question and the answer are the same sentence. That is a definition unfolding, not a coincidence.

4:14And you can see that the word NOUGHT is doing all the work, by changing it. Set the same polynomial equal to ONE instead, and the two sets come apart immediately. That was measured, not asserted. Over thirty-eight thousand values, the equation set equal to nought and the polynomial agreed every single time. Set equal to one, they disagreed in both directions: two hundred and fifty values satisfy the new equation without being zeroes, and two hundred and fifty-eight are zeroes without satisfying it.

4:54Now for the payoff, and it is a large one for so small an observation. You already know something about zeroes: a quadratic polynomial has at most two of them. That is a fact about expressions. It says nothing about equations. But roots and zeroes are the same numbers. So the fact crosses over without any new argument at all. A quadratic equation has at most two roots. Notice what did NOT happen. Nobody solved anything. Nobody factorised anything.

5:30The ceiling on the number of answers arrived before any method for finding them did. That ceiling is worth being careful about, because it is very easy to believe for the wrong reason. Four hundred and eighty-six quadratics were swept, and the numbers at which each of them comes out as nothing were counted. Three hundred and forty-eight of them have none on the grid. Eighteen have exactly one. A hundred and twenty have two.

6:02Not one of them has three. But that on its own proves nothing, because a counter that cannot count past two would give the same answer. So the same counter, unchanged, was put to ten cubics. It said three every time. Which is what makes the quadratic table a measurement rather than a restatement of the thing being measured. There is one case where the counting gets delicate, and it is worth meeting head on.

6:34Take three x squared minus two root six x plus two equals nought. Its root is root two over root three, which is the same number as root six over three. Watch it pass. That number squared is two thirds, so the first term is three times two thirds, which is two. The middle term is minus two root six times root six over three. Root six times root six is six, six over three is two, and twice that is four. So the middle term is minus four.

7:09And the last term is two. Two, minus four, two. Nothing. It passes. Now, that equation factorises into the same linear factor twice. The factor appears twice, so the root is reported twice. So there is ONE distinct number that solves it, and TWO roots. Both statements are true at once, and neither is a slip. The second counts factors; the first counts numbers. One more thing, and it is practical.

7:43Checking a proposed root and finding one are not the same size of job, and it is worth knowing which you are doing. Checking is one substitution. It needs no method, no factorising, and no formula — just arithmetic. Finding needs a method, which is what the rest of this subject supplies. That difference was measured. Checking one candidate took six arithmetic operations. Finding the same roots by trying every value on the grid took four hundred and seventy-four — seventy-nine times as many.

8:21Which is why you are asked to verify at the end of a worked example, and why it is never a waste of time. It is the only step in the whole chain that does not require you to trust the method you just used. A word on the checking, because this is the easiest claim in the subject to confirm without testing anything. 'Root of the equation' and 'zero of the polynomial' are the same condition — and any checker that works both of them out from the same line of code makes that true by construction.

8:59Evaluate once, call the answer 'is a zero', call it again 'is a root', and the identity is proved with no mathematics in it. So the two were settled from different objects. The ROOT question was asked of the equation written as a claim with two sides — two x squared plus one equals three x — by working out the left side, working out the right side, and asking whether they agree. No polynomial is formed at all.

9:31The ZERO question was asked of the single tidied expression, by asking whether its value is nothing. A third route never evaluates anything: it reads the roots off a factorisation by dividing each factor's constant by its coefficient. All three agree everywhere, and a fourth — one that compares the first term with the last, which is a real mistake and not an invented one — is wrong on nine of them.

10:03The surds were done in exact arithmetic throughout. No decimal appears anywhere in the checking, because root six over three is not a decimal. A number is a root of a quadratic equation exactly when putting it in makes the expression come out as nothing. Which is exactly what it means for that number to be a zero of the polynomial. Root, solution, satisfies, zero: four words, one condition, different grammatical homes.

10:38Because they are the same numbers, the ceiling on zeroes becomes a ceiling on roots. At most two, and no method was needed to say so. A repeated factor supplies its root twice, so an equation can have two roots and one distinct answer. And checking is cheap. Do it.