Question

According to the fundamental theorem of algebra, how many roots does the polynomial f(x)=x^4+3x^2+7have, over the complex numbers, and counting roots with multiplicity greater than one as distinct (i.e. f (x)=x^2 has two roots, both are zero).

Answer 1

Step-by-step explanation:

Given According to the fundamental theorem of algebra, how many roots does the polynomial f(x)=x^4+3x^2+7 have, over the complex numbers, and counting roots with multiplicity greater than one as distinct

• Fundamental theorem of algebra is a polynomial of degree n > 0
• Now the equation is f(x) = x^4 + 3x^2 + 7. This has a degree 4.
• Now we have m^2 + 3m + 7
• So x^4 + 3x^2 + 7 = (x^2 – px + √7) (x^2 + px + √7)
•                                 = x^4 + x^2 (2√7 – m^2) + 7
• Now putting 2√7 – p^2 = 3
•    So p^2 = 2√7 – 3
•  So p = √2√7 – 3
•    So x^4 + 3x^2 + 7 = (x^2 - √2√7 – 3x + √7) (x^2 + √2√7 – 3x + √7)

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