SOLUTION: I am having trouble with determining the polynomial function for the problem of two zeros, one is 2 with multiplicity 1 and the other is 1 with multiplicity 3. The polynomial is of

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I am having trouble with determining the polynomial function for the problem of two zeros, one is 2 with multiplicity 1 and the other is 1 with multiplicity 3. The polynomial is of      Log On


   

Question 1052680: I am having trouble with determining the polynomial function for the problem of two zeros, one is 2 with multiplicity 1 and the other is 1 with multiplicity 3. The polynomial is of the fourth degree and and is asking for the leading coefficient to be 1.
I really appreciate any help or insight you could give, as I am fairly lost. Thank you!
Found 3 solutions by Fombitz, htmentor, Alan3354:
Answer by Fombitz(32388) About Me  (Show Source):
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x=2 is a zero so %28x-2%29 is a factor with an exponent of 1.
x=1 is a zero so %28x-1%29 is a factor with an exponent of 3.
f%28x%29=1%2A%28x-1%29%5E3%28x-2%29
f%28x%29=%28x-1%29%5E3%28x-2%29
Expanding,
f%28x%29=x%5E4-5x%5E3%2B9x%5E2-7x%2B2

Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
The multiplicity is determined by the exponent of the factor that produced the root. So the factor for the multiplicity 1 root will be a first order polynomial and the multiplicity 3 root will have a third order polynomial. We are given the two roots and we must have the leading coefficient equal to 1, so we can factor the polynomial in the following way: (x-1)^3(x-2) = 0. This can be further written as (x-1)(x-1)(x-1)(x-2), which makes it easy to see the multiplicity. Setting this equal to zero we see that this gives the correct roots, 1 (mult. 3) and 2 (mult. 1).
Carrying out the multiplication gives f(x)= x^4-5x^3+9x^2-7x+2
The graph is shown below:
+graph%28+500%2C+400%2C+-1%2C+3%2C+-0.5%2C+2%2C+%28x-1%29%5E3%2A%28x-2%29%29+

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
I am having trouble with determining the polynomial function for the problem of two zeros, one is 2 with multiplicity 1 and the other is 1 with multiplicity 3. The polynomial is of the fourth degree and and is asking for the leading coefficient to be 1.
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It's f(x) = (x-2)*(x-1)^3