Chapter 2: Polynomials
In this chapter, we explore the algebraic expressions known as polynomials, focusing on their degree, zeros, and the relationship between coefficients and zeros.
1. Basic Definitions
A polynomial is an expression consisting of variables and coefficients. The highest power of the variable in a polynomial $p(x)$ is called the degree of the polynomial.
- Linear Polynomial: A polynomial of degree 1. General form: $ax + b$.
- Quadratic Polynomial: A polynomial of degree 2. General form: $ax^2 + bx + c$.
- Cubic Polynomial: A polynomial of degree 3. General form: $ax^3 + bx^2 + cx + d$.
2. Geometrical Meaning of the Zeros
The zero of a polynomial $p(x)$ is the $x$-coordinate of the point where the graph of $y = p(x)$ intersects the x-axis.
Note: A polynomial $p(x)$ of degree $n$ has at most $n$ zeros. For a quadratic polynomial, the graph is a curve called a parabola.
3. Relationship between Zeros and Coefficients
For a quadratic polynomial $p(x) = ax^2 + bx + c$, let the two zeros be $\alpha$ and $\beta$.
| Property | Formula |
|---|---|
| Sum of Zeros ($\alpha + \beta$) | $-\frac{b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$ |
| Product of Zeros ($\alpha \cdot \beta$) | $\frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}$ |
If $\alpha$ and $\beta$ are given, the quadratic polynomial can be formed using:
$p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta]$ (where $k$ is a constant)
4. Relationship for Cubic Polynomials
For a cubic polynomial $ax^3 + bx^2 + cx + d$ with zeros $\alpha, \beta, \gamma$:
- $\alpha + \beta + \gamma = -\frac{b}{a}$
- $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$
- $\alpha\beta\gamma = -\frac{d}{a}$
5. Division Algorithm for Polynomials
If $p(x)$ and $g(x)$ are any two polynomials with $g(x) \neq 0$, then we can find polynomials $q(x)$ and $r(x)$ such that:
$p(x) = g(x) \times q(x) + r(x)$
Where $r(x) = 0$ or degree of $r(x) <$ degree of $g(x)$.