Help please! How does the function f(x)=x^2−4x−21 illustrate the Fundamental Theorem of Algebra?

Fill in the blanks. (Hint: Your answers are numbers.)

The degree of f(x) is . The Fundamental Theorem of Algebra guarantees that a polynomial equation has the same number of complex roots as its degree. This means that f(x) has exactly zeros. Those zeros are 7 and .

f(x)=x²−4x−21

The degree is the biggest power of x.  That's a polynomial of degree 2, also called a quadratic function.  Let's find its zeros.

0 = x²−4x−21 = (x - 7)(x+3)

x=7 or x=-3

The fundamental theorem guarantees every non-constant polynomial with complex coefficients has a complex zero, let's call it r.  If we divide the polynomial by x-r there won't be any remainder and we'll get a new polynomial, one degree less.  The fundamental theorem again applies and (if it's not a constant polynomial) we are assured of another zero, s.  We divide by x-s and get a new polynomial of degree one less.  We repeat all this until we get a constant polynomial (degree zero).    So we get a zero for every degree. They're not necessarily all different.

The degree of f(x) is 2. The Fundamental Theorem of Algebra guarantees that a polynomial equation has the same number of complex roots as its degree. This means that f(x) has exactly 2 zeros. Those zeros are 7 and -3.

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