63-66= Finding a Polynomial with Specified Zeros Find a polynomial of the specified degree that

63-66= Finding a Polynomial with Specified Zeros Find a...

Key Concepts

Zeros of a Polynomial

Zeros, or roots, are the values of the variable for which the polynomial equals zero. In constructing a polynomial from given zeros, each zero corresponds to a factor of the form (x - zero), and the product of these factors, possibly multiplied by a leading coefficient, forms the polynomial.

Factor Theorem

The Factor Theorem states that a number r is a root of a polynomial if and only if (x - r) is a factor of that polynomial. This theorem is essential for building a polynomial when its zeros are known, as it provides a direct method to translate each zero into a factor.

Degree of the Polynomial

The degree of a polynomial is determined by the highest power of the variable in its expanded form. When constructing a polynomial from its zeros, the degree is typically the sum of the multiplicities of the zeros. For distinct zeros with multiplicity one, the number of factors equals the degree.

Constructing a Polynomial from Given Zeros

To construct a polynomial from its given zeros, one writes it as a product of linear factors corresponding to each zero. If the zeros are given as r1, r2, r3, etc., the polynomial can be expressed as P(x) = a·(x - r1)(x - r2)(x - r3)..., where a is a nonzero constant that can be chosen based on any additional requirements.

Transcript

00:01 All right, just like the last one, this does not have to be too difficult.

00:03 If you know that negative 2 is 0, well, you know that you can factor out on x plus 2, because if i plug in negative 2, that'll be 0.

00:12 If negative 1 is a factor, then i can put x plus 1, because if i plug in negative 1, it'll become 0.

00:18 I could write x plus 0 or x minus 0, but i also could just write x.

00:25 Because if 1x is 0, whole thing's 0.

00:28 All right now we're at x minus 1 and x minus 2 would suffice to plug in 1 and 2 all right now we just got to do some distribution why don't we go ahead and do this first x times x is x squared 2 times x is 2x 1 times x is 1x so that gives me a 3x and 1 times 2 is 2 and i'm going to do something a little bit tricky here let's just go ahead and take this x and distribute it all the way over here.

00:59 So instead of this being x squared, i'm going to say, well, if i multiply another x in there, it's an x to the third.

01:06 If i multiply another an x in here, it's not just x, it's x squared.

01:10 And that is now a 2x.

01:12 Again, i just took this x and distributed it to all my terms.

01:17 Okay, x times x is x squared over here.

01:22 Negative 1 times x is negative 1x.

01:24 Negative 2 times x is negative 2x.

01:27 Combined is a negative 3x.

01:29 And negative 1 times negative 2 positive 2 wow gonna have quite a bit to distribute here we'll do it one at a time how about this i'll do blue for multiplying x squared times everything x squared times x to the third is x to the fifth x squared times 3x squared is 3x to the fourth x squared times 2x is 2x to the third all right let's switch colors let's go green for negative 3x negative 3x times x x x third is negative 3x to the 4th.

02:07 The exponents are very important here.

02:10 Negative 3x times 3x squared is negative 9x to the third.

02:18 Negative 3x times a positive 2x is a negative 6x squared.

02:25 And we finished with that term.

02:27 So let's do a red for this.

02:28 2 times x to the third is 2x to the 3rd.

02:33 2 times 3x squared.

02:34 2 times 3x squared...

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