The polynomial P(x) = 5x^2(x - 2)^3(x + 6) has degree . It...

The polynomial P(x) = 5x^2(x - 2)^3(x + 6) has degree ____. It has zeros 0, 2, and ____. The zero 0 has multiplicity ____, and the zero 2 has multiplicity ____. (a) If a is a zero of the polynomial P(x), then ____ must be a factor of P(x). (b) If a is a zero of multiplicity m of the polynomial P(x), then ____ must be a factor of P(x) when we factor P completely.

Transcript

00:04 So in this problem, it says you've got this polynomial.

00:08 It says has degree of blank.

00:11 And then it has zero, zero two in blank.

00:17 Essentially to figure out the degree of the polynomial, what do you want to know is what all your x's together end up being.

00:27 So if you have an x squared and then an x cubed from the second part of that, and then an x from this part of it, and the x cubed comes from there.

00:40 If you multiply that all together, that would be x to the sixth.

00:44 So that means this would be a sixth power.

00:48 And so that polynomial is to degree six or sixth power.

00:54 And it says it has zeros of zero two and what? well, if you were going to solve this to get it zeros, you would say 5x squared equals zero, and you would say x minus two, equals 0 and you'd say x plus 6 equals 0 then you'd solve each one of those so you at this first one that if i've squared equal 0 you're going to divide by 5 and so you'd have x squared equal 0 or if you score it both sides then x equals 0 on the second one you'd add 2 to each side so you'd get x equals 2 and on the last then you'd subtract 6 from both sides and you get x equals negative 6 and so you'll notice in your thing, it says it has zeros, zero, which is one of the ones we found, two, which is another one we found, and the last one would be this negative six.

01:53 Then it says the zero has multiplicity of blank.

01:57 Well, the way you check the multiplicity is look at the powers.

02:01 So that's an x squared, which we also saw it down here.

02:05 So since it's an x squared, you would have multiplicity of two.

02:12 And then the 0, 2 has multiplicity of 3, because that x minus 2 was cubed.

02:23 And then your multiplicity on the 6 would just be a 1.

02:27 So it's only their 1 time that assumed the 1...

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