Introduction to Polynomials

A polynomial is an expression made up by adding and subtracting monomials [1 2]. Also the concepts of the degree and the leading coefficient of a polynomial are introduced. A polynomial in standard form as well constant, linear, quadratic, and cubic polynomials are also defined. Examples, questions and their solutions are presented.

Monomials

We start by defining a monomial as a term of the form \[ \large \color{red}{ a x^n } \] where \( x \) is a variable, \( a \) is a constant and \( n \) is a nonnegative integer.

Examples of Monomials

1. \( \quad 2 x^2 \)
2. \( \quad - 3 x \)
3. \( \quad \dfrac{1}{2} x^7 \)

Binomials

We now define a binomial as a sum/difference of 2 monomials that are not like.

Examples of Binomials

1. \( \quad 2 x + 8 \)
   
2. \(\quad - x^3 + 3 x \)
3. \(\quad \dfrac{1}{2} x^2 - x \)

Trinomials

A trinomial as a sum/difference of 3 monomials that are not like.

Examples of Trinomials

1. \( \quad 2 x^3 + 8 x - 2 \)
2. \( \quad -\dfrac{1}{6} x^4 - 5 x - 9 \)
3. \( \quad 0.2 x^2 - x + 4 \)

Polynomials

A polynomial in \( x \) is the sum of any number of monomials and has the following form
\[ \large \color{red}{ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_1 x + a_0 } \]
where the coefficients \( a_k \) are constant. If coefficient \( a_n \) is not equal to \( 0 \), then \( n \) (the highest power) is the degree of the polynomial and \( a_n \) is the leading coefficient.

Examples of Polynomials

Polynomials in Standard Form

A polynomial is in standard form when it is written such that the power of the variable is in descending order.

Example

1. \( \quad 4x - 9 + x^2 \)
2. \( \quad 4x^3 - 0.03x^4 - x^2 \)
3. \( \quad 4 - 3x^5 + x^2 - \dfrac{1}{2}x^3 \)
4. \( \quad 3x^2 - 3 - \sqrt{3}\,x^3 \)

Solution

1. \( \quad x^2 + 4x - 9 \) — degree \(2\), leading coefficient \(1\)
2. \( \quad -0.03x^4 + 4x^3 - x^2 \) — degree \(4\), leading coefficient \( -0.03 \)
3. \( \quad -3x^5 - \frac{1}{2}x^3 + x^2 + 4 \) — degree \(5\), leading coefficient \( -3 \)
4. \( \quad -\sqrt{3}\,x^3 + 3x^2 - 3 \) — degree \(3\), leading coefficient \( -\sqrt{3} \)

Equal Polynomials

Two polynomials are equal if their corresponding coefficients are all equal.

Example

Solution

Example

Solution

Constant Polynomials

A constant polynomial has a degree equal to \( 0 \) and is therefore of the form \( b \), where \( b \) is any contant.

Example

Linear Polynomials

A linear polynomial has a degree equal to \( 1 \) and is therefore of the form \( a x + b \)

Example

Quadratic Polynomials

A quadratic polynomial has a degree equal to \( 2 \) and is therefore of the form \( a x^2 + b x + c \)

Example

Cubic Polynomials

A cubic polynomial has a degree equal to \( 3 \) and is therefore of the form \( a x^3 + b x^2 + cx + d \)

Example

Questions

Part A

Classify the following as monomials , binomials or trinomials .
\( 2 x^2 \)   ,   \( - x^2 - 2 x^3 \)   ,   \( -6 \)   ,   \( - \dfrac{1}{5} x^2 + 4 x^5 - 9 \)   ,   \( \sqrt 5 x^2 - x - 9 \)   ,   \( \dfrac{1}{2} x^2 - x^3 + 4\)   ,   \( x^3 + 4\)

Part B

Write in standard form and determine the degree and the leading coefficient of each polynomial.

1. \( \quad -x^3 + 2x^2 - 1 \)
2. \( \quad -x^2 - \dfrac{1}{2}x^3 \)
3. \( \quad -0.0001x^2 - 9 \)
4. \( \quad -x - 9 - \dfrac{\sqrt{2}}{2}x^2 \)
5. \( \quad -\dfrac{1}{3}x^2 - x^3 - \dfrac{1}{5}x^4 \)

Part C

Find, if possible, the unknown coefficients so that the polynomials are equal

1. \( \quad x^2 + 2x - 1 = ax^2 + bx + c \)
2. \( \quad -x^3 + 2x^2 - 1 = ax^3 + bx^2 + cx + d \)
3. \( \quad - x^3 + 2x -\dfrac{1}{2}x^4 = ax^4 + bx^3 + cx^2 + dx + e \)
4. \( \quad - 2x^2 - x -x^5 + 9 = ax^4 + bx^3 + cx^2 + dx + e \)

Part D

Classify the following polynomials as constant, linear, quadratic, or cubic.

1. \( \quad -6x^2 - 2x - 1 \)
2. \( \quad -1 \)
3. \( \quad -\sqrt{5}\,x^3 + 2x^2 - 1 \)
4. \( \quad -5 + 2x \)
5. \( \quad -x - 2x^3 - x^2 + 9 \)
6. \( \quad -\dfrac{x^2}{3} \)
7. \( \quad -100 \)
8. \( \quad -x^2 + 9 + \sqrt{3}\,x^3 \)
9. \( \quad -x \)

Solutions to the Above Questions

Part A

Part B

1. \( \quad -x^3 + 2x^2 - 1 \)   — degree \(3\), leading coefficient \( -1 \)
2. \( \quad -\dfrac{1}{2}x^3 - x^2 \)   — degree \(3\), leading coefficient \( -\dfrac{1}{2} \)
3. \( \quad -0.0001x^2 - 9 \)   — degree \(2\), leading coefficient \( -0.0001 \)
4. \( \quad -\dfrac{\sqrt{2}}{2}x^2 - x - 9 \)   — degree \(2\), leading coefficient \( -\dfrac{\sqrt{2}}{2} \)
5. \( \quad -\dfrac{1}{5}x^4 - x^3 - \dfrac{1}{3}x^2 \)   — degree \(4\), leading coefficient \( -\dfrac{1}{5} \)

Part C

1. Both the left and right polynomials are written in standard form and include all terms.

   \[ x^2 + 2x - 1 = ax^2 + bx + c \]

   Identifying the coefficients gives: \( a = 1,\; b = 2,\; c = -1 \).

2. Both polynomials are written in standard form. Rewrite the left polynomial including missing terms using zeros:

   \[ -x^3 + 2x^2 + 0x - 1 = ax^3 + bx^2 + cx + d \]

   Identifying the coefficients gives: \( a = -1,\; b = 2,\; c = 0,\; d = -1 \).

3. Rewrite the polynomial on the left in standard form, including missing terms using zeros:

   \[ -\frac{1}{2}x^4 - x^3 + 0x^2 + 2x + 0 = ax^4 + bx^3 + cx^2 + dx + e \]

   Identifying the coefficients gives: \( a = -\frac{1}{2},\; b = -1,\; c = 0,\; d = 2,\; e = 0 \).

4. The given polynomials do not have the same degree and therefore cannot be equal.

Part D

1. \( \quad -6x^2 - 2x - 1 \): degree \(2\), therefore a quadratic polynomial.
2. \( \quad -1 \): degree \(0\), therefore a constant polynomial.
3. \( \quad -\sqrt{5}x^3 + 2x^2 - 1 \): degree \(3\), therefore a cubic polynomial.
4. \( \quad -5 + 2x \): degree \(1\), therefore a linear polynomial.
5. \( \quad -x - 2x^3 - x^2 + 9 \): degree \(3\), therefore a cubic polynomial.
6. \( \quad -\dfrac{x^2}{3} \): degree \(2\), therefore a quadratic polynomial.
7. \( \quad -100 \): degree \(0\), therefore a constant polynomial.
8. \( \quad -x^2 + 9 + \sqrt{3}x^3 \): degree \(3\), therefore a cubic polynomial.
9. \( \quad -x \): degree \(1\), therefore a linear polynomial.

More References and Links to Polynomial Functions