FUNCTIONS 06
Polynomial Functions
Polynomial functions combine non-negative integer powers of a variable. Their graphs are smooth and their overall shape is strongly influenced by the highest-power term.
The main idea
A polynomial function has the form
\[\boxed{f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0},\]where \(n\) is a non-negative integer and \(a_n\ne0\). Linear and quadratic functions are therefore special cases of polynomial functions.
1. Degree describes the highest power
The degree is the largest exponent with a non-zero coefficient.
For
\[f(x)=4x^5-2x^2+7,\]the degree is \(5\).
\(-2x^2\)
\(+7\)
degree = \(5\)
2. Coefficients and missing terms
In
\[f(x)=3x^3-5x+2,\]the coefficient of \(x^2\) is \(0\), even though the term is not written.
\[3x^3+0x^2-5x+2.\]coefficient of \(x^2\): \(0\)
coefficient of \(x\): \(-5\)
constant: \(2\)
3. The constant term gives the y-intercept
At \(x=0\), every term containing \(x\) disappears, leaving
\[f(0)=a_0.\]So the constant term is also the vertical intercept.
4. Roots are x-axis crossings or touches
A root is a value \(r\) satisfying
\[f(r)=0.\]For
\[f(x)=x^2-5x+6=(x-2)(x-3),\]the roots are \(2\) and \(3\).
5. Factors and roots are two views of the same idea
If \(f(r)=0\), then \(x-r\) is a factor of \(f(x)\).
\[f(r)=0\quad\Longleftrightarrow\quad(x-r)\text{ is a factor}.\]↓
\(f(r)=0\)
↓
factor \((x-r)\)
6. Odd and even degree give different end behaviour
For large \(|x|\), the highest-power term dominates.
An even-degree polynomial with positive leading coefficient rises at both ends. An odd-degree polynomial with positive leading coefficient falls to the left and rises to the right.
7. Cubic functions can have two turning points
A degree-3 polynomial can bend twice. For example, a cubic may have one local maximum and one local minimum.
8. Degree limits the number of turning points
A polynomial of degree \(n\) can have at most \(n-1\) turning points.
This is an upper limit, not a guarantee.
degree 2 → at most 1 turn
degree 3 → at most 2 turns
degree 4 → at most 3 turns
9. Repeated roots change how the graph meets the axis
For a simple root, the graph normally crosses the \(x\)-axis. For an even repeated root, the graph can touch the axis and turn back.
10. Polynomials are smooth everywhere
Polynomial graphs have no breaks, jumps, vertical asymptotes or sharp corners. They are continuous and differentiable for every real \(x\).
11. Polynomials are easy to differentiate term by term
For
\[f(x)=3x^4-2x^2+5x-7,\]the derivative is
\[f'(x)=12x^3-4x+5.\]Each power is handled separately using the power rule.
\(-2x^2\to-4x\)
\(5x\to5\)
\(-7\to0\)
12. Polynomial approximation
Polynomials are also important because many smooth functions can be approximated locally by polynomial expressions.
This idea later leads to Taylor polynomials and Taylor series.
↓ near one point
polynomial approximation
↓
easier analysis
A polynomial is built from non-negative integer powers. Its degree controls much of its overall shape, roots correspond to factors, the leading term determines end behaviour, and polynomial graphs are smooth for all real inputs.
Why this matters for calculus
Polynomials are central examples in differentiation and integration because the power rule applies term by term. They also provide local approximations to much more complicated smooth functions.