Polynomial Functions w/ zeros.

In summary, the conversation discusses finding a polynomial function with given zeros and the possibility of simplifying the factors. The suggested method is to use the monic polynomial of lowest degree, which is x^2-8x+11 for the given zeros.
  • #1
AznBoi
471
0
Ok I have a probelm with find the polynoimal function which has these zeros:

Zeros: 2, 4+sqrt.(5), 4-sqrt.(5)

Find the polynomial equation with the given zeros.

So far I know:

y=(x-2)(x-(4+sqrt.(5))(x-(4-sqrt.(5))

but is there any way I could make the factor (x-(4+sqrt.(5)) into a better one? For example:

Zeros: -2,-1,0,1,2

I did:
y=x(x^2-1)(x^2-4)

instead of y=x(x+2)(x+1)(x-1)(x-2)

Thanks! :biggrin:
 
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  • #2
AznBoi said:
Ok I have a probelm with find the polynoimal function which has these zeros:
Zeros: 2, 4+sqrt.(5), 4-sqrt.(5)

Find the polynomial equation with the given zeros.
Not "the" polynomial function- "a" polynomial function. There are an infinite number of polynomial functions having these zeros.

So far I know:

y=(x-2)(x-(4+sqrt.(5))(x-(4-sqrt.(5))

but is there any way I could make the factor (x-(4+sqrt.(5)) into a better one?
Yes, much as you did with -2, 2, and -1, 1 below: [itex](x-(4+\sqrt{5}))(x- (4-\sqrt{5}))= ((x-4)+\sqrt{5})((x-4)-\sqrt{5})= (x-4)^2- 5[/itex]
[itex]= x^2- 8x+ 16- 5= x^2- 8x+ 11 [/itex]
That is the monic polynomial of lowest degree having those roots.
For example:

Zeros: -2,-1,0,1,2

I did:
y=x(x^2-1)(x^2-4)

instead of y=x(x+2)(x+1)(x-1)(x-2)

Thanks! :biggrin:
 
  • #3


I would say that your approach is correct. You have correctly identified the zeros and created a polynomial equation using the factors. In terms of making the factor (x-(4+sqrt.(5)) into a better one, it is not necessary since you have already found a valid solution. However, if you want to simplify the equation, you could try expanding the factors and combining like terms to see if you can simplify it further. Otherwise, your approach is perfectly acceptable.
 

1. What are polynomial functions with zeros?

Polynomial functions with zeros are mathematical functions that can be written in the form of f(x) = anxn + an-1xn-1 + ... + a1x + a0, where a represents constants and n is a positive integer. These functions have one or more solutions, or zeros, where the graph of the function crosses the x-axis.

2. How do you find the zeros of a polynomial function?

The zeros of a polynomial function can be found by setting the function equal to zero and solving for the value(s) of x. This can be done by using factoring, the quadratic formula, or other algebraic methods. Graphing the function can also help to visually identify the zeros.

3. What is the degree of a polynomial function with zeros?

The degree of a polynomial function is the highest power of x in the function. For example, a polynomial function with the form f(x) = 3x4 - 2x2 + 5 has a degree of 4. The number of zeros in a polynomial function cannot exceed its degree.

4. Can a polynomial function have complex zeros?

Yes, a polynomial function can have complex zeros. Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i2 = -1). Complex zeros come in conjugate pairs, meaning that if a + bi is a zero, then a - bi is also a zero.

5. How do polynomial functions with zeros relate to the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that a polynomial function of degree n has n complex zeros, counting multiplicities. This means that a polynomial function with degree 4 can have up to 4 complex zeros. The theorem helps to explain the behavior of polynomial functions and their zeros, and it is an important concept in the study of polynomial functions.

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