In this section, we define the zero of a polynomial p(x) as a value 'c' such that p(c) = 0. The section begins by evaluating a polynomial at various points, demonstrating how to compute p(x) for specific values to find its zeroes. Key examples, such as determining whether specific numbers are zeroes of given polynomials, illustrate the concept clearly. Furthermore, it discusses the unique properties of linear polynomials and their zeroes, emphasizing that every linear polynomial has exactly one zero, while non-zero constant polynomials have none. The zero polynomial, by convention, has all real numbers as zeroes. The section concludes with several exercises designed to reinforce understanding of finding and verifying zeroes of polynomials.
Example:
Check whether \(-1\) and \(3\) are zeros of the polynomial \(x^2 - 2x - 3\).
Solution: Let \( p(x) = x^2 - 2x - 3 \).
Then
\[ p(-1) = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0 \]
\[ p(3) = (3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0 \]
Therefore, \(-1\) is a zero of the polynomial \(x^2 - 2x - 3\), and \(3\) is also a zero.