Polynomials

Polynomials are algebraic expressions made up of constants and in determinate. They are used to represent numbers in almost all branches of mathematics. For instance, 2x + 9 and x² + 3x + 11 are polynomials.

The highest exponent is the polynomial's degree. Polynomials can be subjected to a variety of operations, including addition, subtraction, multiplication, and division.
 

Examples of Polynomials

2x + 9

This is a polynomial of degree 1 (linear polynomial).

\(x2 + 3x + 11\)

This is a polynomial of degree 2 (quadratic polynomial).

\(5x3 − 4x + 7\)

This is a polynomial of degree 3 (cubic polynomial).

7

This is a constant polynomial with degree 0.

\(​​​​​​​4x4 + x2 − 6\)

This is a polynomial of degree 4.

The degree of a polynomial is the highest or greatest exponent of the variable in the polynomial. 
This degree is used in Descartes’ rule of signs for calculating the maximum zeroes a polynomial equation can have.

Let us take an example.

Example:The polynomial 3x⁴ + 7 has a degree of 4.

Here, the degree of the polynomial is the highest exponent amongst all variables.
 

The standard form of a polynomial is the way of writing all terms in descending order. The expression with the highest exponent comes first, then the terms with descending powers. The process ends at the constant term, which has no variables.
 

For example: Arrange the polynomial 4 + 3x² + x in standard form.

Explanation: To express the above expression in standard form, we will find the highest exponent in this expression, which is x², so the 3x² term will come first.

Then, we will arrange them in decreasing powers accordingly until the constant term at the end. So, the standard form of 4 + 3x² + x will be: 3x² + x + 4. 

Polynomials are classified into different types based on their degree and the number of terms. Based on the number of terms, polynomials are classified as monomials, binomials, and trinomials.
 

• A monomial is an expression that consists of only one term, such as 5x or −3a², 7y, etc. 
   
• A binomial contains two terms, such as x + 4 or 3y − 2. 
   
• A trinomial comprises three terms, such as x² + 3x + 2. 

Polynomials can be classified based on their degree (the highest power of the variable).
 

• A constant polynomial has a degree of zero, like 7.
   
• A linear polynomial has a degree of one, such as 2x + 5.
   
• Polynomials with the highest degree of 2 are called quadratic polynomials, for example, x² − 4x + 1.
   
• While a polynomial with a degree of 3, like x³ + 2x² − x + 6  is a cubic polynomial.

These classifications make it easier to identify and solve polynomials.

A term in a polynomial is a single component of the expression that consists of a variable raised to a power (called an exponent) and a number (called a coefficient). Positive (+) or negative (-) symbols are used to separate terms.

Every polynomial is composed of one or more terms. Among the terms are

• A variable (e.g., x, y)
   
• An exponent, which is a whole number.
   
• Coefficients are numbers attached to variables, while constants are numbers without variables.

Theorem 1: Degree of Polynomials

Definition: For two polynomials A and B:

The degree of (A ± B) is less than or equal to the greater of the degrees of A and B.

The degree of (A × B) is the sum of their degrees.

Example

A(x) = 3x² + 2x

B(x) = x³ − 5

Degree of (A + B) = 3

Degree of (A × B) = 2 + 3 = 5

Theorem 2: Division Algorithm

Definition: For any polynomials A and B, where B ≠ 0, there exist polynomials Q and R such that
A = BQ + R, where the degree of R is less than the degree of B.

Example

x² + 3x + 2 ÷ (x + 1)
= (x + 1)(x + 2) + 0

Theorem 3: Factor (Bézout’s) Theorem

Definition: (x − a) is a factor of P(x) if and only if P(a) = 0.

Example

P(x) = x² − 4
P(2) = 0

So, (x − 2) is a factor.

Theorem 4: Zeros of Factors

Definition: If polynomial Q divides polynomial P, then every zero of Q is also a zero of P.

Example

P(x) = (x − 1)(x + 2)

Zeros are 1 and −2.

Theorem 5: Factorization Theorem

Definition: A polynomial of degree n can be expressed as a product of n linear factors.

Example

2x² − 8 = 2(x − 2)(x + 2)

Theorem 6: Number of Roots

Definition: A polynomial of degree n has exactly n roots, counting multiplicities.

Example

x³ − 1 has three roots.

Theorem 7: Coprime Divisibility

Definition: If a polynomial is divisible by two coprime polynomials, then it is divisible by their product.

Example

If a polynomial is divisible by (x − 1) and (x + 1),

then it is divisible by (x − 1)(x + 1).

Theorem 8: Complex Conjugate Roots

Definition: If a real polynomial has a complex root, then its complex conjugate is also a root.

Example

If 2 + 3i is a root, then 2 − 3i is also a root.

Theorem 9: Factorization of Real Polynomials

Definition: A real polynomial can be factored into linear factors and quadratic factors with no real roots.

Example

P(x) = (x − 1)(x² + 4)

Theorem 10: Remainder Theorem

Definition: The remainder when f(x) is divided by (x − a) is f(a).

Example

f(x) = x³ − 2x + 1

Remainder when divided by (x − 2) = f(2) = 5

Theorem 11: Rational Root Theorem

Definition: Any rational root of a polynomial is of the form p/q,

where p divides the constant term, and q divides the leading coefficient.

Example

For 2x³ − 3x − 1,

Possible rational roots are ±1 and ±1/2

Like numbers, polynomials can be used in various mathematical operations, such as addition, subtraction, multiplication, and division. In order to properly simplify or solve polynomial expressions, these operations adhere to certain procedures and guidelines.

1. Addition: Like terms, or terms with the same variable raised to the same power, are combined when adding polynomials.

For example, in the expression (3𝑥² + 2𝑥 + 5) + (4𝑥² − 𝑥 + 1), we add the like terms by combining the coefficients of x^{​​​​2}, x and the constant terms separately: 

3x² + 4x² = 7x^{​​​​​​2}
2x - x = x
5 + 1 = 6
7𝑥² + 𝑥 + 6.

2. Subtraction: Similar to addition, subtraction involves first changing the signs of the terms in the polynomial being subtracted.

For example, in (5𝑥² + 3x − 2) − (2x² + x + 4), we will distribute the minus sign to all terms of the second polynomial. Therefore, (2x²) becomes (−2x²), x becomes −x, and + 4 becomes −4.

− 2x² + x + 4  = − 2x² − x − 4.
= 5𝑥² + 3𝑥 − 2 − 2x² − 𝑥 − 4
= (5x² − 2x²) + (3x − x) (− 2 − 4)
= 3x² + 2x − 6

Changing the signs guarantees accurate subtraction and helps to prevent errors.

3. Multiplication: For the multiplication of polynomials, all terms of the first polynomial are multiplied by all terms in the second. We can use the distributive or FOIL method to multiply 

\((x + 2)(x + 3) = x × x + x × 3 + 2 × x  + 2 × 3\)
=\( x^2 + 3x + 2x + 6 = x^2 + 5x + 6\).  

The answer is \( x^2 + 5x + 6\).

When working with polynomials that contain more than two terms, this process may need several steps.

3. Division: The final operation is polynomial division, a more complex process, and the most commonly used methods are synthetic division or long division. On occasion, the expression can be simplified by factoring instead of performing long division.

For instance, the expression \(x^2 + 3𝑥 + 2\) can be easily divided by 𝑥 + 1 by first factoring the numerator into (𝑥 + 1) (𝑥 + 2), now the expression can be simplified into x + 2 after canceling out the common factor.

Factorization of polynomials means breaking a polynomial into simpler factors that, when multiplied, give the original expression. For example,\( x² + 5x + 6\) can be written as (x + 2)(x + 3). It helps in finding roots, simplifying expressions, and solving equations using methods like grouping, common factors, or special formulas such as the difference of squares.

Polynomial Equations

A polynomial equation is an equation formed by setting a polynomial equal to zero.
The general form of a polynomial equation is

\(P(x) = anxn + an-1xn-1 + … + a1x + a0 = 0,\)

where an, an-1, …, a0 are constants and an is not zero.

Solving a polynomial equation means finding the value or values of the variable that make the equation true.

Examples

\(x2 + 3x + 2 = 0\)

\(x3 + x + 1 = 0\)

\(x + 7 = 0\)

Polynomial Functions

A polynomial function is a function defined by a polynomial expression. It contains variables with non-negative integer powers, along with constants and coefficients.

General form

f(x) = anxn + an-1xn-1 + … + a1x + a0, where an ≠ 0.

Examples

\(f(x) = x2 + 4\)

\(g(x) = −2x3 + x − 7\)

\(g(x) = −2x3 + x − 7\)

Finding the values of the variable (often 𝑥) that cause the equation to equal zero is the process of solving polynomial equations. We refer to these values as roots or solutions. This is a detailed tutorial on how to solve polynomial equations:

• Setting the polynomial to zero is always the first step in solving a polynomial equation. By doing this, we may solve for the values of the variable (often 𝑥) that make the expression zero by converting the function into an equation. 

For example, you would rewrite a polynomial like \(x^2 − 5x + 6\) as x^{​​​​​2} − 5x + 6 = 0 if you were given it. The next step is to identify the values of 𝑥 for which this equality is valid; these are referred to as the equation's roots or solutions.

• Factoring is a widely used and simple method for solving polynomial equations. The polynomial is rewritten using this method as a product of simpler expressions, most frequently binomials.

𝑥² − 5x + 6 can be factored into (x − 2) (x − 3), for instance, meaning that x = 2 and x = 3 are the solutions since such numbers make each component zero. Simple polynomials, particularly quadratics or ones with recurring patterns like perfect square trinomials or the difference of squares, are good candidates for factoring.

• The quadratic formula can be used when factoring is challenging or unclear, particularly when dealing with quadratics. The following is the formula:
   

                                                                \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Any quadratic equation with the form 𝑎𝑥² + bx + c = 0 can be solved using this formula. After entering the values for 𝑎, 𝑏, and 𝑐, you can simplify. It is especially helpful when the polynomial has complicated or irrational roots or doesn't factor smoothly. For instance, the quadratic formula yields the solutions x = 1 and x = −3 for the equation 𝑥² + 2x − 3 = 0. 

•  For polynomials of degree three or higher, methods such as synthetic division or long division are applied when we know at least one root. By dividing out a known factor, these techniques enable you to lower the polynomial's degree and gradually simplify the equation until it is simpler to solve. The Rational Root Theorem, which proposes potential rational solutions by dividing factors of the constant term by factors of the leading coefficient, might be used if you are unsure of any roots. 

After that, you test these values to check if they add up to zero using synthetic division or substitution.

•  Graphing is another method used for solving polynomial equations. It helps us visualize the roots as points of intersection of curves on the x-axis. All intercepts on the x-axis give us the real roots of a polynomial.

The graph for the polynomial f(x) = x³ − 4x² − 7x + 10 is shown below. The real roots of the equation are represented by the spots where the curve crosses the x-axis. This visual method aids in locating approximations of solutions and comprehending the behavior of the function. 

The zeros of polynomial functions are the values of the variable that make the function zero. In other words, the values of x when f(x) = 0 are the zeros of a polynomial function f(x). 

The values of the variable that make a polynomial equal to zero are called zeros, also known as roots or solutions. In other words, the values of x when f(x) = 0 are the zeros of a polynomial function f(x). The values of variables are important for evaluating polynomial functions. Let \(f(x) = x² − 5x + 6\) be the polynomial, after factorization, (x - 2) (x - 3) = 0, therefore, x = 2,3. 

Determining these values is essential to comprehending and evaluating polynomial functions, particularly in the context of graphing or equation solving.

Let's look at a basic example to better grasp this idea. Let \(f(x) = x² − 5x + 6\) be the polynomial in question. We set it to zero to determine its zeros: \(x² − 5x + 6 = 0\). The expression can be factored to obtain (x − 2)(x − 3) = 0. This indicates that x = 2 and x = 3 are the polynomial's zeros because entering either number into the function yields zero.

In a graphic representation, a polynomial's zeros match the graph's x-intercepts. These are the locations on the x-axis where the curve meets or crosses. For example, the graph will intersect the x-axis three times in a polynomial with three real zeros. Only the real zeros will show up as intercepts on the graph if some zeros are complex (using imaginary values).

Three distinct polynomial functions are displayed in this graph to represent different kinds of zeros:

• The green curve represents the function \(𝑓(𝑥) = 𝑥^2 − 5x + 6\), where the real zeros are at x = 2 and x = 3.

• The blue curve, \(f(x) = (x − 4)^2\), exhibits a "bounce” at the x-axis and a recurrent real zero at x = 4.

• The red curve,\( f(x) = x² + 4\), does not intersect the x-axis because it has no real zeros; its roots are complex numbers.

Mastering polynomials becomes easier with a clear understanding of their structure and properties. Here are some quick tips to simplify solving and manipulating polynomial expressions effectively.

• Learn to identify coefficients, variables, and degrees before solving problems.
   
• Always add or subtract terms with the same variable and exponent.
   

• Students should understand variables, coefficients, constants, exponents, and degrees.
   

• Begin with linear (degree 1) and quadratic (degree 2) polynomials before higher degrees

• Apply polynomial concepts to real-life situations like area or profit calculations for better understanding.

Polynomials are vital tools in research, technology, and daily decision-making because they are used to model motion, forecast trends, and address real-world issues in disciplines including engineering, physics, economics, and medicine.

• Construction and engineering - In engineering and construction, polynomial equations are used to model physical systems and structures. For instance, computations based on polynomial expressions are frequently used in the design of bridges, buildings, and tunnels. To model stress, load distribution, or arch forms, civil engineers employ quadratic or cubic functions. In order to ensure stability and safety, polynomials are used to forecast how materials will react to forces.

• Economics and business - In the field of economics or business, polynomials are used for representing profit function, revenue and cost. A polynomial function is used to predict how the cost changes over time or increase in production cost. Even polynomial curves are used for predicting trends, profit points, marginal costs. 

• Animation and computer graphics - In computer graphics and animation, polynomials are used for drawing curves and shapes. Polynomial functions are used to create Bézier curves, which are used to create scalable typefaces and graphics. Game developers and animators use polynomial interpolation to generate realistic motion, effects, and transitions. 

• Biology and medicine - For medicine dosage, spread of any disease, or growth curves of any biological objects, polynomial functions are used. For forecasting variations in drug, or population growth over time, or any kind of new medicine formulation, polynomial equations are used. 

• Space exploration and navigation - Additionally, polynomial functions are essential for space exploration and GPS navigation. They simulate the paths of spacecraft, satellites, and space probes. Scientists guarantee precise navigation and timing, whether it's for landing on Mars or orbiting the Earth, by computing routes using polynomial equations.