Degree: The highest power of the variable \(x\) in a polynomial \(p(x)\).
Types of Polynomials:
- Linear: Degree \(1\) (e.g., \(2x + 3\)).
- Quadratic: Degree \(2\) (e.g., \(ax^2 + bx + c\)).
- Cubic: Degree \(3\) (e.g., \(ax^3 + bx^2 + cx + d\)).
- Value and Zeros: The value of \(p(x)\) at \(x = k\) is \(p(k)\). If \(p(k) = 0\), then \(k\) is called the zero of the polynomial.
Geometrical Meaning of Zeros:
The zero(s) of a polynomial are the \(x\)-coordinates of the points where its graph intersects the \(x\)-axis.
- Linear: A straight line; intersects the \(x\)-axis at exactly one point: \((\frac{-b}{a}, 0)\).
- Quadratic: A parabolic curve. It can have at most two zeros (it may have \(2\), \(1\), or \(0\) real zeros depending on how many times it touches or crosses the \(x\)-axis).
- Cubic: A curve that can intersect the \(x\)-axis at at most three points.
Relationship Between Zeros and Coefficients
Linear Polynomial (\(ax + b\))
- Zero (\(\alpha\)): \(\alpha = \frac{-b}{a} = \frac{-(\text{Constant term})}{\text{Coefficient of } x}\)
Quadratic Polynomial (\(ax^2 + bx + c\)):
If \(\alpha\) and \(\beta\) are the zeros:
- Sum of zeros: \(\alpha + \beta = \frac{-b}{a}\)
- Product of zeros: \(\alpha\beta = \frac{c}{a}\)
- Formation: \(p(x) = x^2 - (\text{Sum of zeros})x + (\text{Product of zeros})\)
Cubic Polynomial (\(ax^3 + bx^2 + cx + d\))
If \(\alpha, \beta, \text{ and } \gamma\) are the zeros:
- Sum of zeros: \(\alpha + \beta + \gamma = \frac{-b}{a}\)
- Sum of product of zeros (taken two at a time): \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
- Product of zeros: \(\alpha\beta\gamma = \frac{-d}{a}\)
Important!
In general, the graph of the polynomial of \(n\) degree intersects the \(x\) − axis at atmost \(n\) points.
In other words, the polynomial of \(n\) degree has atmost \(n\) zeroes.
In other words, the polynomial of \(n\) degree has atmost \(n\) zeroes.