What is the standard form of a polynomial function with given roots and degree?

In summary, the given roots are 2, -2i, and the degree of the polynomial is 4. The polynomial can be written in standard form as (x-2)(x+2)(x+2i)(x-2i). If the polynomial has real coefficients, then the fourth root could be any real number, resulting in an infinite number of solutions.
  • #1
thomasrules
243
0
The Roots and degree of a polynomial function are given. Write the function in standard form.

b) 2, -2i, degree 4

obviously i know there is a function with [tex]x^4[/tex] and it should have 4 x answers so I don't know how to do this...I know that (x-2) is a factor
 
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  • #2
then what about (x+2i) and who says that 2 can only be a root once? What about -2i?
 
  • #3
Does the polynomial have to be real? If so, note the roots must come in complex conjugate pairs. If not, just multiply together the right number of factors of (x-r), where r is a root.
 
  • #4
yes it has to be real
 
  • #5
then statusX is saying if -2i is a root, then 2i must be a root as well, since 2i is the complex conjugate of -2i [the complex conjugent of a+bi is a-bi, you just have no a in this situation]
 
  • #6
SO IT'S

[tex](x-2)(x+2)(x+2i)(x-2i)[/tex]
 
  • #7
I am confused. The problem says "The Roots and degree of a polynomial function are given" and I would interpret "The Roots" to mean that the the polynomial has only those roots. However, then it couldn't have real coefficients.

If the polynomial has real coefficients and then the roots given are not all. Clearly -2i must be another but the fourth could be any real number. There are an infinite number of solutions: (x-a)(x-2)(x2+4) where a is any real number.
 

What are roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero.

Why are the roots of a polynomial important?

The roots of a polynomial provide information about the behavior and characteristics of the polynomial function, such as its intercepts, turning points, and end behavior.

How do you find the roots of a polynomial?

The roots of a polynomial can be found by factoring the polynomial, using the quadratic formula, or by using synthetic division and the rational root theorem.

What is the difference between real and complex roots of a polynomial?

Real roots of a polynomial are the values of the variable that result in real numbers when substituted into the polynomial. Complex roots are values that result in complex numbers, which include a real and imaginary component, when substituted into the polynomial.

How many roots can a polynomial have?

The number of roots a polynomial can have depends on its degree. A polynomial of degree n can have at most n distinct roots, including any repeated roots.

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