Zeros of polynomials: matching equation to zeros
About this video
Given that a polynomial P has zeros at x = -4, x = 3 and x = 1/8, Sal builds a matching polynomial from factors that are zero at each value: (x + 4)(x - 3)(x - 1/8). He uses these factors to rule out answer choices, and shows that multiplying by a constant like 8 gives (x + 4)(x - 3)(8x - 1) without changing the zeros.
You'll be able to
- write a factor that is zero at a given x-value, such as x + 4 for x = -4.
- build a possible polynomial equation from a list of zeros.
- explain why multiplying a polynomial by a constant does not change its zeros.
Key words
- Zero
- An x-value where the polynomial equals zero.
- Factor
- One of the expressions multiplied together to make the polynomial.
- Constant
- A fixed number, like 8, that you can multiply by without changing where the polynomial is zero.
Transcript
- A polynomial P has zeros when X is equal to negative four, X is equal to three, and X is equal to one-eighth. What could be the equation of P? So pause this video and think about it on your own before we work through it together. All right. So the fact that we have zeros at these values, that means that P of X, when X is equal to one of these values is equal to zero.
So P of negative four is equal to zero, P of three is equal to zero, and P of one-eighth is equal to zero. And before I even look at these choices, I could think about constructing a polynomial for which that is true. That's going to be true if I can express this polynomial as the product of expressions where each of these would make each of those expressions equal to zero.
So what's an expression that would be zero when X is equal to negative four? Well the expression X plus four, this is equal to zero when X is equal to negative four, so I like that. What would be an expression that would be equal to zero when X is equal to three?
Well what about the expression X minus three? If X is equal to three, then this is going to equal to be equal zero. Zero times anything is going to be equal to zero. So P of three would be zero in this case. And then what is an expression that would be equal to zero when X is equal to one-eighth?
Well that would be X minus one-eighth. Now tho-- these aren't the only expressions. You could multiply them by constants and still the principles that I just talked about would be true. But our polynomial would look something like this. You could try it out. If X is equal to negative fou-- (chuckles) if X is equal to negative four, well then this first expression is zero.
Zero times something times something is zero. Same thing for X equals three. If this right over-- If X equals three, then X minus three is equal to zero, and then zero times something times something is zero. And then if X is equal to one-eighth, this expression's going to be equal to zero.
Zero times something times something is going to be equal to zero. So which of these choices look like that? So let's see. X plus four, I actually see that in choices B, and I see that in choices D. Choice C has X minus four there. So that would have a zero at X equals four.
If X equals four, this first-- this first expression-- this first part of the expression would be equal to zero. But we care about that happening when X is equal to negative four. So I would actually rule out C, and then for the same reason I would rule out A.
So we're between B and D, and now let's see. Which of these have an X minus three in them. Well I see an X minus three here. I see an X minus three there. So I like the-- I still like B and D. I'll put another check mark right over there. And then last but not least, which of these would be equal to zero when X is equal to one-eighth?
Well, let's see. If I do one-eighth times one-eighth here I'm gonna get one-sixty-fourth for this part of the expression. And so that's not going to be equal to zero. And these other two things aren't going to be equal to zero when X is equal to one-eighth, so this one is not looking so good, but let's verify this one.
This has-- If X is equal to one-eighth, we have eight times one-eighth which is one, minus one. That is going to be equal to zero, so this one checks out. And you might be thinking hey! This last polynomial looks a little bit different than this polynomial that I wrote up here when I just tried to come up with a polynomial for which this would be true.
And as I mentioned, you could take this and multiply it by constants and it would still be true. So if you just take this, and if we were to multiply it by eight, you would get P of X down here, because if we were to multiply this times eight, which wouldn't change the zeros, well then if you distribute this eight, this last expression would become eight X minus one.
Which is exactly what we have down here.
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