4 Polynomial Functions
Definition 4.1. A polynomial function is one that can be written \[P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0,\qquad a_n\neq 0,\] where \(n\) is a non-negative integer called the degree. The numbers \(a_0,\ldots ,a_n\) are the coefficients, \(a_n\) is the leading coefficient and \(a_0\) the constant term.
| Degree | Form | Name | Graph |
| \(0\) | \(P(x)=a\) | constant | horizontal line |
| \(1\) | \(P(x)=ax+b\) | linear | straight line of gradient \(a\) |
| \(2\) | \(P(x)=ax^2+bx+c\) | quadratic | parabola, one turn |
| \(3\) | \(P(x)=ax^3+bx^2+cx+d\) | cubic | two turns or none |
Note 4.2. Every polynomial function \(P(x)\) has a matching equation \(P(x)=0\). A number \(c\) with \(P(c)=0\) is called a zero of the function, or a root of the equation. The two words describe the same number from two points of view.
4.1 Linear equations
4.1.1 Equations with fractions
4.1.2 Cross-multiplying
4.2 Quadratic functions and equations
4.2.1 Completing the square
4.2.2 The quadratic formula and the nature of the roots
4.2.3 An application
4.3 Division of polynomials
4.3.1 The remainder and factor theorems
4.4 Rational functions
4.5 Partial fractions
4.6 Inequalities
4.7 The modulus function
4.8 Practice problems
4.1.1 Equations with fractions
4.1.2 Cross-multiplying
4.2 Quadratic functions and equations
4.2.1 Completing the square
4.2.2 The quadratic formula and the nature of the roots
4.2.3 An application
4.3 Division of polynomials
4.3.1 The remainder and factor theorems
4.4 Rational functions
4.5 Partial fractions
4.6 Inequalities
4.7 The modulus function
4.8 Practice problems
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