Polynomial Functions | Elective Mathematics

Definition and Types

Lesson Video

Watch this comprehensive explanation on definition and types of polynomial functions.

A polynomial function is an algebraic expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.

General form of a polynomial:

\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]

Where $ a_n \neq 0 $ and $ n $ is a non-negative integer.

Degree of Polynomial:

The highest power of the variable (n) determines the degree:

Constant: Degree 0 (e.g., $ P(x) = 5 $)
Linear: Degree 1 (e.g., $ P(x) = 2x + 3 $)
Quadratic: Degree 2 (e.g., $ P(x) = x^2 - 5x + 6 $)
Cubic: Degree 3 (e.g., $ P(x) = x^3 + 2x^2 - x + 5 $)

Leading Coefficient:

The leading coefficient of a polynomial is the term with the highest degree ($ a_n $).
The leading coefficient cannot be zero ($ a_n \ne 0 $)

Linear Polynomials

Linear Polynomials are first-degree polynomials of the form below:

\[ f(x) = ax + b \quad (a \neq 0) \]

Characteristics of linear Polynomials:

Straight line graphs
One real root (x-intercept)
Constant slope (gradient)

Graph of $ f(x) = 2x - 1 $ with the root at $ x = 0.5 $

Quadratic Polynomials

Second-degree polynomials are of the form:

\[ f(x) = ax^2 + bx + c \quad (a \neq 0) \]

Graph Characteristics:

Parabolic shape
Vertex (turning point)
Axis of symmetry
0, 1, or 2 real roots

Graph of $ f(x) = x^2 -5x + 6 $ with the root at $ x = 2 $ and $ x = 3 $

Cubic Functions

Third-degree polynomials of the form:

\[ f(x) = ax^3 + bx^2 + cx + d \quad (a \neq 0) \]

Example cubic function:

\[ f(x) = x^3 - 3x^2 - x + 3 \]

Graph Characteristics:

"S"-shaped curve
1 to 3 real roots
Possible 1 or 2 turning points

Graph of $ f(x) = x^3 - 3x^2 - x + 3 $ with the roots at $ x = -1 $, $ x = 1 $ and $ x = 3 $

Practice Exercise

Question 1

Which of the following is a polynomial function?

$ f(x) = 3x^2 - 5x + 7 $
$ f(x) = \frac{2}{x} + 1 $
$ f(x) = \sqrt{x} + 4 $
$ f(x) = x^{-2} + 3x $

Answer: A

$ f(x) = 3x^2 - 5x + 7 $ is a polynomial because its variable has only non-negative integer exponents.

The other expressions contain a negative exponent, a variable in the denominator, or a fractional exponent.

Question 2

Consider the polynomial \[ P(x) = 4x^5 - 3x^3 + 7x^2 - 2x + 9. \] State:

the degree of the polynomial;
the leading coefficient;
the constant term.

The polynomial is \[ P(x) = 4x^5 - 3x^3 + 7x^2 - 2x + 9. \]

(a) Degree: $5$, because the highest power of $x$ is $5$.

(b) Leading coefficient: $4$, because $4x^5$ is the term with the highest degree.

(c) Constant term: $9$.

Question 3

Classify each of the following polynomial functions as constant, linear, quadratic, or cubic.

$ f(x) = 8 $
$ g(x) = 5x - 2 $
$ h(x) = 2x^2 + 3x - 1 $
$ p(x) = x^3 - 4x + 6 $

(a) $ f(x) = 8 $ — Constant polynomial (degree $0$).

(b) $ g(x) = 5x - 2 $ — Linear polynomial (degree $1$).

(c) $ h(x) = 2x^2 + 3x - 1 $ — Quadratic polynomial (degree $2$).

(d) $ p(x) = x^3 - 4x + 6 $ — Cubic polynomial (degree $3$).

Question 4

Consider the polynomial \[ f(x) = -2x^3 + 5x^2 - 7x + 4. \] Which statement about the polynomial is correct?

It is a quadratic polynomial with leading coefficient $5$.
It is a cubic polynomial with leading coefficient $-2$.
It is a cubic polynomial with leading coefficient $4$.
It is a linear polynomial with leading coefficient $-2$.

Answer: B

The highest power of $x$ is $3$, so the polynomial is cubic.

The term containing the highest power is $-2x^3$, so the leading coefficient is $-2$.

Question 5

Match each polynomial type with its correct graph characteristic.

Linear polynomial
Quadratic polynomial
Cubic polynomial

Characteristics:

Parabolic shape with a possible turning point
S-shaped curve with up to two turning points
Straight-line graph with constant gradient

Answers:

(a) Linear polynomial → (iii)
A linear polynomial has a straight-line graph with constant gradient.

(b) Quadratic polynomial → (i)
A quadratic polynomial has a parabolic graph and may have a turning point.

(c) Cubic polynomial → (ii)
A cubic polynomial generally has an S-shaped graph and may have one or two turning points.

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Full Practice Set

Roots and Zeros

Lesson Video

Watch this comprehensive explanation on roots and zeros of polynomial functions.

Roots and Zeros

A root (or zero) of a polynomial, $p(x)$, is a solution to $ p(x) = 0 $.

For $ p(x) = x^2 - 4 $:

$ \begin{aligned} & \Rightarrow p(x) &= 0 \\ & \Rightarrow x^2 - 4 &= 0 \\ & \Rightarrow (x - 2)(x + 2) &= 0 \end{aligned} $

if $ x - 2 = 0$
$\Rightarrow x = 2$

if $ x + 2 = 0$
$\Rightarrow x = -2$

Thus, $ x = -2 $ and $ x = 2 $ are roots or zeros of the polynomial.

Fundamental Theorem of Algebra:

A polynomial of degree $ n $ has exactly $ n $ roots (real or complex, counting multiplicities).

Practice Exercise

Question 1

Find the zeros of the polynomial $f(x) = 2x^2 + 3x - 2$.

Question 2

Find the roots of the polynomial $g(x) = x^2 - 5x + 6$.

Question 3

Solve for the zeros of $h(x) = x^2 - 9$.

Question 4

Determine the x-intercepts of the polynomial $p(x) = x^2 + 2x - 8$.

Question 5

Find all roots of $q(x) = x^2 - 7x + 12$.

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Full Practice Set

Addition & Subtraction

Lesson Video

Watch this comprehensive explanation on addtion and subtraction of polynomial functions.

Addition and Subtraction

Combine like terms (terms with the same degree):

Example:

\[ (3x^2 + 2x - 5) + (2x^2 - 4x + 1) = 5x^2 - 2x - 4 \]

Practice Exercise

Question 1

Given $h(x) = x^4 + 2x^3 - x^2 + 4x - 7$ and $g(x) = 2x^3 + 3x^2 - 5x - 5$, find $h(x) + 2g(x)$.

Question 2

Given $h(x) = x^4 + 2x^3 - x^2 + 4x - 7$ and $g(x) = 2x^3 + 3x^2 - 5x - 5$, find $h(x) - 2g(x)$.

Question 3

Given $h(x) = x^4 + 2x^3 - x^2 + 4x - 7$ and $g(x) = 2x^3 + 3x^2 - 5x - 5$, find $3h(x) - 2g(x)$.

Question 4

Given $h(x) = x^4 + 2x^3 - x^2 + 4x - 7$ and $g(x) = 2x^3 + 3x^2 - 5x - 5$, find $g(x) - h(x)$.

Question 5

Given $h(x) = x^4 + 2x^3 - x^2 + 4x - 7$ and $g(x) = 2x^3 + 3x^2 - 5x - 5$, find $2h(x) - 2g(x)$.

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Full Practice Set

Multiplication

Lesson Video

Watch this comprehensive explanation on multiplication of polynomial functions.

Multiplication

Use the distributive property (FOIL method for binomials):

Example:

\[ (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]

Practice Exercise

Question 1

Multiply and simplify: $ (x + 2)(x^2 - 3x + 4) $.

Question 2

Multiply and simplify: $ (x - 3)(x^2 + 2x + 5) $.

Expand:

$ x(x^2 + 2x + 5) - 3(x^2 + 2x + 5) = x^3 + 2x^2 + 5x - 3x^2 - 6x - 15 $

Simplify:

$ x^3 - x^2 - x - 15 $

Question 3

Multiply and simplify: $ (2x + 1)(x^2 - x + 4) $.

Expand:

$ 2x(x^2 - x + 4) + 1(x^2 - x + 4) = 2x^3 - 2x^2 + 8x + x^2 - x + 4 $

Simplify:

$ 2x^3 - x^2 + 7x + 4 $

Question 4

Multiply and simplify: $ (x^2 + 3x - 2)(x + 4) $.

Expand:

$ x^2(x + 4) + 3x(x + 4) - 2(x + 4) = x^3 + 4x^2 + 3x^2 + 12x - 2x - 8 $

Simplify:

$ x^3 + 7x^2 + 10x - 8 $

Question 5

Multiply and simplify: $ (3x - 2)(2x^2 + x - 5) $.

Expand:

$ 3x(2x^2 + x - 5) - 2(2x^2 + x - 5) = 6x^3 + 3x^2 - 15x - 4x^2 - 2x + 10 $

Simplify:

$ 6x^3 - x^2 - 17x + 10 $

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Full Practice Set

Division of Polynomials

Lesson Video

Watch this comprehensive explanation on division of polynomial functions.

Division of Polynomials

Polynomial long division follows similar steps to numerical division:

Divide $ x^3 - 2x^2 - 4x + 8 $ by $ x - 2 $:

1. Divide leading terms: $ x^3 ÷ x = x^2 $

2. Multiply divisor by $ x^2 $: $ x^3 - 2x^2 $

3. Subtract and bring down next term

4. Repeat process until remainder is of lower degree than divisor

Result: $ x^2 - 4 $ with remainder 0