Question
How many roots, real or complex, does the polynomial 7 + 5x* - 3x2 have in all?
Answers
GPT-4o mini
To determine the number of roots of the polynomial \(7 + 5x - 3x^2\), we first observe that it is a quadratic polynomial because the highest power of \(x\) is 2.
A general quadratic polynomial is of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constant coefficients. In this case, we can write the polynomial as:
\[
-3x^2 + 5x + 7
\]
Here, \(a = -3\), \(b = 5\), and \(c = 7\).
According to the Fundamental Theorem of Algebra, any polynomial of degree \(n\) has exactly \(n\) roots in the complex number system (counting multiplicities). Since this is a quadratic polynomial (\(n = 2\)), it will have exactly 2 roots in total. These roots can either be real or complex.
To summarize, the polynomial \(7 + 5x - 3x^2\) has exactly **2 roots** in total.
A general quadratic polynomial is of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constant coefficients. In this case, we can write the polynomial as:
\[
-3x^2 + 5x + 7
\]
Here, \(a = -3\), \(b = 5\), and \(c = 7\).
According to the Fundamental Theorem of Algebra, any polynomial of degree \(n\) has exactly \(n\) roots in the complex number system (counting multiplicities). Since this is a quadratic polynomial (\(n = 2\)), it will have exactly 2 roots in total. These roots can either be real or complex.
To summarize, the polynomial \(7 + 5x - 3x^2\) has exactly **2 roots** in total.