6.2.2 Fundamental Theorem of Algebra Key Objectives • Use the Fundamental Theorem of Algebra and its corollary to write a
polynomial equation of least degree with given roots. • Identify all of the roots of a polynomial equation.
The following statements about real roots of polynomial equations are equivalent. • A real number r is a root of the polynomial equation P ( x ) = 0. • P ( r ) = 0 • r is an x -intercept of the graph of P ( x ). • x − r is a factor of P ( x ). • When you divide the rule for P ( x ) by x − r , the remainder is 0. • r is a zero of P ( x ). Example 1 Writing Polynomial Functions Given Zeros
To write the simplest polynomial function given its zeros, use the fact that if r is a zero of P ( x ), then x − r is a factor of P ( x ) to write the zeros as binomial factors of P ( x ). Then multiply the factors to write the function in expanded form.
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