2.11 Polynomial Functions (HL)

1. Definition and Terminology

A polynomial function, or simply a polynomial, is an algebraic expression defined in the following form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$

Where $a_n \ne 0$, all coefficients $a_i \in \mathbb{R}$, and the exponent $n \in \mathbb{N}$.

The highest power of $x$ is defined as the degree of the polynomial, denoted as $\deg f(x) = n$.
For example, if $f(x) = 5x^4 + 3x^2 - 7x + 2$, then $\deg f(x) = 4$.
If $g(x) = x^5 - 2x^3 + 5x - 7$, then $\deg g(x) = 5$.

Standard Polynomial Terminology

Degree	Standard Form	Classification
$\deg f(x) = 0$	$f(x) = a$	Constant Function
$\deg f(x) = 1$	$f(x) = ax + b$	Linear Function
$\deg f(x) = 2$	$f(x) = ax^2 + bx + c$	Quadratic Function
$\deg f(x) = 3$	$f(x) = ax^3 + bx^2 + cx + d$	Cubic Function
$\deg f(x) = 4$	$f(x) = ax^4 + bx^3 + cx^2 + dx + e$	Quartic Function

Note: The degree of the zero polynomial $f(x) = 0$ is undefined (or defined as $-1$ or $-\infty$ in advanced texts).

2. Addition and Multiplication of Polynomials

Operations on polynomials inherently produce another polynomial. The operations follow standard algebraic expansion and grouping techniques.

EXAMPLE 1

Let $f(x) = 3x^2 - 2x + 5$ and $g(x) = 2x^3 - 7x + 1$.

Addition:
$f(x) + g(x) = (3x^2 - 2x + 5) + (2x^3 - 7x + 1) = \mathbf{2x^3 + 3x^2 - 9x + 6}$.

Multiplication:
$f(x)g(x) = (3x^2 - 2x + 5)(2x^3 - 7x + 1)$
$= 6x^5 - 21x^3 + 3x^2 - 4x^4 + 14x^2 - 2x + 10x^3 - 35x + 5$
$= \mathbf{6x^5 - 4x^4 - 11x^3 + 17x^2 - 37x + 5}$.

Degree Rules for Operations

If $\deg f(x) = n$ and $\deg g(x) = m$:

If $n > m$, then $\deg[f(x) + g(x)] = n$.
If $n = m$, then $\deg[f(x) + g(x)] \le n$ (leading coefficients might cancel out).
For multiplication, $\deg[f(x)g(x)] = n + m$.
For self-multiplication, $\deg[f(x)f(x)] = 2n$.

3. Division of Polynomials

The division of two polynomials $f(x)$ by $g(x)$ yields a quotient polynomial $q(x)$ and a remainder polynomial $r(x)$ such that:

$f(x) = g(x)q(x) + r(x)$

Where $r(x) = 0$ or $\deg r(x) < \deg g(x)$. This can also be expressed fractionally as $\dfrac{f(x)}{g(x)} = q(x) + \dfrac{r(x)}{g(x)}$. If $r(x) = 0$, $g(x)$ is a strict factor of $f(x)$.

EXAMPLE 2 (Polynomial Long Division)

Divide $f(x) = 2x^3 - 4x^2 + 5x - 1$ by $g(x) = x^2 + 3x + 1$.

Standard Layout (Right-to-Left)

               2x - 10
        ___________________
x²+3x+1 | 2x³ - 4x² + 5x - 1
        -(2x³ + 6x² + 2x)
          --------------
              -10x² + 3x - 1
            -(-10x² -30x -10)
              --------------
                     33x + 9

Alternative Layout (Left-to-Right)

 2x³ - 4x² + 5x - 1 | x² + 3x + 1
                    |_____________
-(2x³ + 6x² + 2x)   | 2x - 10
-----------------   |
     -10x² + 3x - 1 |
   -(-10x² -30x -10)|
     -------------- |
            33x + 9 |

Quotient: $q(x) = 2x - 10$

Remainder: $r(x) = 33x + 9$

Result: $2x^3 - 4x^2 + 5x - 1 = (x^2 + 3x + 1)(2x - 10) + (33x + 9)$

EXAMPLE 3

Divide $f(x) = 2x^3 + 2x^2 - x - 1$ by $g(x) = 2x^2 - 1$.

Standard Layout (Right-to-Left)

               x + 1
        ___________________
2x² - 1 | 2x³ + 2x² - x - 1
        -(2x³       - x)
          -----------------
                2x²     - 1
              -(2x²     - 1)
                -----------
                          0

Alternative Layout (Left-to-Right)

2x³ + 2x² - x - 1 | 2x² - 1
                  |________
-(2x³       - x)  | x + 1
----------------  |
        2x²     - 1 |
      -(2x²     - 1)|
        ----------- |
                  0 |

Quotient: $q(x) = x + 1$

Remainder: $r(x) = 0$

Therefore, $2x^3 + 2x^2 - x - 1 = (2x^2 - 1)(x + 1)$.
Conclusion: The divisor $(2x^2 - 1)$ is a perfect factor of $f(x)$.

4. The Factor and Remainder Theorems

The Factor Theorem:
A polynomial $f(x)$ is perfectly divisible by $(x-a)$ if and only if $f(a) = 0$.

Proof: If divisible, $f(x) = (x-a)q(x)$, rendering $f(a) = 0$. Conversely, if $f(x) = (x-a)q(x) + r$, and $f(a) = 0$, then $r = 0$.

The Remainder Theorem:
When a polynomial $f(x)$ is divided by a linear factor $(x-a)$, the remainder is strictly equal to the scalar value $f(a)$.

Proof: Using the identity $f(x) = (x-a)q(x) + r$. Evaluating at $x=a$ eliminates the quotient term $(a-a)q(a) = 0$, leaving exactly $f(a) = r$.

EXAMPLE 4

Let $f(x) = x^3 + x^2 - x + 2$. Evaluate the remainders when divided by specific linear factors.

Divided by $(x-1)$: Evaluate $f(1) = 3$. The remainder is 3.
Divided by $(x+1)$: Evaluate $f(-1) = 3$. The remainder is 3.
Divided by $(x-2)$: Evaluate $f(2) = 12$. The remainder is 12.
Divided by $(x+2)$: Evaluate $f(-2) = 0$. The remainder is 0, making $(x+2)$ an exact factor.

EXAMPLE 5

Find $c$ such that $x-2$ is a factor of $f(x) = x^3 - cx^2 + 5x - 2$.

According to the Factor Theorem, we must have $f(2) = 0$.
$f(2) = 2^3 - c(2)^2 + 5(2) - 2 = 0 \Rightarrow 8 - 4c + 10 - 2 = 0 \Rightarrow 16 - 4c = 0$
Therefore, $\mathbf{c = 4}$.

EXAMPLE 6 (Solving Cubic Equations)

Solve $x^3 + x^2 - x + 2 = 0$.

From Example 4, we established that $x=-2$ is a root, meaning $(x+2)$ is a factor.
Dividing the cubic polynomial by $(x+2)$ yields the quotient $(x^2 - x + 1)$.
The factored form is $(x+2)(x^2 - x + 1) = 0$.
The quadratic component $x^2 - x + 1 = 0$ has no real roots because its discriminant is negative ($\Delta = (-1)^2 - 4(1)(1) = -3 < 0$).
Thus, the only real root is $\mathbf{x = -2}$.

EXAMPLE 7 (Solving Cubic Equations)

Solve $f(x) = x^3 - 6x^2 + 11x - 6 = 0$.

Testing simple integers reveals $f(1) = 1 - 6 + 11 - 6 = 0$, thus $(x-1)$ is a factor.
Long division of $f(x)$ by $(x-1)$ produces the quadratic quotient $(x^2 - 5x + 6)$.
The equation becomes $(x-1)(x^2 - 5x + 6) = 0$.
Factoring the quadratic gives $(x-1)(x-2)(x-3) = 0$.
The solutions are strictly $\mathbf{1, 2, \text{ and } 3}$.

Remark on Guessing Roots

For a general polynomial $f(x) = a_n x^n + \dots + a_0$, potential rational roots are discovered by testing fractions formed by:

$\pm \dfrac{\text{Factors of } a_0}{\text{Factors of } a_n}$

If $a_n = 1$, potential integer roots correspond exclusively to the integer factors of the constant term $a_0$.

EXAMPLE 8

Find the roots of $f(x) = 2x^3 - 7x^2 - 17x + 10$.

Potential integer roots are derived from factors of 10: $\pm 1, \pm 2, \pm 5, \pm 10$.

Potential rational roots include fractions divided by factors of 2: $\pm \dfrac{1}{2}, \pm \dfrac{5}{2}$.

Testing reveals $f(-2) = 0$, $f(5) = 0$, and $f(1/2) = 0$.

Alternatively, establishing the first root $x = -2$ allows division by $(x+2)$, which yields the remaining quadratic factor $2x^2 - 11x + 5$. This factors cleanly into $(2x - 1)(x - 5)$, confirming the roots are $\mathbf{-2, 5, \text{ and } \dfrac{1}{2}}$.

5. The Graph of a Cubic Function

For a cubic function $f(x) = ax^3 + bx^2 + cx + d$, the leading coefficient $a$ strictly dictates the extreme boundary behavior:

If $\mathbf{a > 0}$, as $x \rightarrow +\infty$, $f(x) \rightarrow +\infty$ (ending upwards). As $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$.
If $\mathbf{a < 0}$, as $x \rightarrow +\infty$, $f(x) \rightarrow -\infty$ (ending downwards). As $x \rightarrow -\infty$, $f(x) \rightarrow +\infty$.
The roots dictate structural intersections on the x-axis. A single root crosses straight through, a squared root $(x-r_1)^2$ tangentially bounces off the axis, and a cubed root flattens while crossing.

THE GRAPH OF A CUBIC FUNCTION

Consider a cubic function

$f(x)=ax^3+bx^2+cx+d$

The leading coefficient $a$ determines the behavior of the graph towards the right end:

If $a>0$, then for large values of $x$, $f(x)\to+\infty$.
If $a<0$, then for large values of $x$, $f(x)\to-\infty$.

The factorization of the cubic function determines the position of the graph in relation to the $x$-axis:

$f(x)$	$a>0$	$a<0$
$a(x-r_1)(x-r_2)(x-r_3)$
$a(x-r_1)^2(x-r_2)$
$a(x-r_1)^3$
$a(x-r_1)(x^2+px+q)$ irreducible

A factor $(x-r)$ of odd multiplicity makes the graph cross the $x$-axis at $x=r$.
A factor $(x-r)^2$ of even multiplicity makes the graph touch or bounce off the $x$-axis at $x=r$.
A factor $(x-r)^3$ makes the graph cross with flattening at $x=r$.
An irreducible quadratic factor $x^2+px+q$ contributes no additional real roots.