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What is the statement of the Fund Theorem of Algebra?
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Equivalently, the field of complex numbers is algebraically closed. Sometimes, this theorem is stated as: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity. Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.
That's a fancy way of saying a first-order polynomial has 1 root a second-order polynomial has 2 roots a third-order polynomial has 3 roots etc. Given all that then, what order is your polynomial and how many roots must it have?
f(x) = 15x^17 + 41x^12 + 13x^3 – 10 The order of a polynomial in one variable x is the highest power of x in the polynomial. For instance x - 2 is first order, because the highest power of x is x^1 = x x^2 - x - 2 is second order, because the highest power of x is x^2 15x^3 + 15x - 27 is third order, because the highest power of x is x^3 Hence given your polynomial 15x^17 + 41x^12 + 13x^3 – 10 what is its order?
i have no clue
What's the higher power of x in f(x) = 15x^17 + 41x^12 + 13x^3 – 10 There are three choices: x^17 x^12 x^3
Right. The highest power of x is x^17 ...
hence f(x) is a 17th-order polynomial ...
hence by the Fundamental Theorem of Algebra f(x) = 15x^17 + 41x^12 + 13x^3 – 10 has 17 roots