Chapter 2: Polynomials
Types of Polynomials and Their Zeroes
Polynomials are expressions in variables with whole number powers. Their degree determines their type: linear (degree 1), quadratic (degree 2), and cubic (degree 3). The zeroes of a polynomial are the values of the variable for which the polynomial equals zero. Graphically, these zeroes are the x-coordinates where the graph intersects the x-axis. A linear polynomial has exactly one zero; a quadratic can have two, one, or none depending on whether its graph (a parabola) cuts the x-axis in two, one, or zero points; a cubic polynomial can have up to three zeroes.
Relationship Between Zeroes and Coefficients
There’s a direct relationship between the zeroes of a polynomial and its coefficients. For a quadratic polynomial ax²+bx+c, the sum of the zeroes is −b/a and the product is c/a. Similarly, for a cubic polynomial ax3+bx2+cx+dax3+bx2+cx+d, the sum of the zeroes is −b/a, the sum of the product of zeroes taken two at a time is c/a, and the product of the zeroes is −d/a. These relationships help construct or verify polynomials when zeroes are known.
Graphical Representation and Applications
The chapter emphasizes the geometric meaning of zeroes. Linear polynomials are straight lines crossing the x-axis once. Quadratic polynomials are parabolas that may intersect the x-axis at two, one, or zero points. Cubic polynomials may intersect it up to three times. Through worked examples and factorization techniques, students learn how to determine the zeroes algebraically and visually, and how to create polynomials with given zeroes. This understanding forms the basis for solving equations and modeling real-world situations.