Polynomial Equation

What Is a Polynomial Equation?

A polynomial equation is an equation that sets a polynomial equal to zero. At its core, a polynomial is an expression made up of variables (like $x$ or $y$), coefficients (the numbers multiplied by the variables), and exponents that are non-negative integers ($0,1,2,3,...$). When you take an expression like this and set it equal to $0$, you create a puzzle to solve: what value(s) of the variable make the statement true?

The general form of a polynomial is $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_2{x}^2+a_{1}x+a_0$. The $a$ values are the coefficients, $x$ is the variable, and the $n$ values are the exponents. To make it an equation, we simply add an equals sign:

$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$

Here are some key characteristics:

• Variables: The letters in the equation, like $x$.
• Coefficients: The numbers in front of the variables. In $3x^2-7x+2=0$, the coefficients are $3$, $-7$, and $2$.
• Exponents: The powers the variables are raised to. They must be whole numbers ($0,1,2,...$).
• Terms: The parts of the polynomial separated by plus or minus signs, like $3x^2$, $-7x$, and $2$.

Examples of polynomial equations:

• $5x-10=0$
• $x^2+4x+3=0$
• $y^3-8=0$

NOT polynomial equations:

• $3x^{-2}+2x-1=0$ (because of the negative exponent)
• $4\sqrt{x}+7=0$ (because $\sqrt{x}$ is the same as $x^{1/2}$, which is a fractional exponent)
• $\frac{2}{x+1}-5=0$ (because this involves division by a variable)

How Do You Classify Polynomials by Degree and Terms?

Polynomials are classified in two primary ways: by their degree and by their number of terms. Understanding these labels helps you predict how an equation will behave and what methods you might use to solve it.

The degree of a polynomial is the value of its highest exponent. This is the most important classification because it tells you the maximum number of solutions (or 'roots') the equation can have.

The number of terms is simply how many pieces are being added or subtracted. A polynomial with one term is a monomial, two terms is a binomial, and three terms is a trinomial.

Here’s a handy table to help you classify them:

For example, the polynomial equation $4x^3-2x+11=0$ is a cubic trinomial. Its degree is $3$ (from the $x^3$ term), meaning it can have up to three distinct solutions, and it has three terms ($4x^3$, $-2x$, and $11$).

What Are Standard Form and 'Roots' of an Equation?

Before you can solve most polynomial equations, you need to arrange them into standard form. Standard form makes it easier to identify coefficients and apply solving techniques like factoring or the quadratic formula.

An equation is in standard form when:

1. All terms are on one side of the equals sign, and the other side is $0$.
2. The terms are written in descending order of their exponents, from highest to lowest.

For example, to write $3x+5=-2x^2$ in standard form, you would move all terms to the left side and order them: $2x^2+3x+5=0$.

Once an equation is in standard form, your goal is to find its roots. The words roots, solutions, and zeros all mean the same thing: they are the specific values of the variable that make the equation true. For the equation $x-4=0$, the root is $x=4$ because $4-4=0$.

A huge idea in algebra, the Fundamental Theorem of Algebra, tells us something powerful about roots. In simple terms, it states that the degree of a polynomial equation tells you the maximum number of roots it can have. A quadratic (degree 2) equation will have at most two real roots, a cubic (degree 3) will have at most three real roots, and so on. This helps you know how many solutions you should be looking for.

How Do You Solve Simple Linear Equations?

Linear equations are polynomials of degree 1. They have the form $ax+b=0$. These are the most straightforward polynomial equations to solve because your goal is simply to isolate the variable (get $x$ by itself) using inverse operations.

The process is usually two steps:

1. Add or subtract the constant term from both sides of the equation.
2. Multiply or divide to remove the coefficient in front of the variable.

Let's walk through an example.

What Are the Main Ways to Solve Quadratic Equations?

Quadratic equations (degree 2) are a major focus in algebra. They are written in the standard form $a{x}^2+bx+c=0$, where $a$ is not zero. Because of the $x^2$ term, we can't just isolate $x$ like we do in linear equations. Instead, we use more advanced methods.

There are three primary methods for solving quadratics:

1. Factoring: This method involves rewriting the quadratic as a product of two linear binomials. It relies on the Zero Product Property, which states that if $A\cdot B=0$, then either $A=0$ or $B=0$ (or both). Factoring is fast and efficient, but it only works when the quadratic is easily factorable.
2. Completing the Square: This is a powerful technique that can solve any quadratic equation by turning one side of it into a perfect square trinomial. It is often used to derive the quadratic formula itself.
3. The Quadratic Formula: This is the universal solver. It works for every quadratic equation, whether it factors or not. All you need to do is identify the coefficients $a$, $b$, and $c$ from the standard form and plug them into the formula.

The $\pm$ symbol means you will perform the calculation once with a plus sign and once with a minus sign, which is how you get the two potential solutions.

How Can Factoring Help Solve Higher-Degree Equations?

For polynomial equations with a degree of 3 or higher, like cubics ($x^3$) and quartics ($x^4$), there are no simple formulas like the quadratic formula. At this level, the primary method for solving them is factoring.

The strategy is to break down the complex polynomial into a product of simpler linear or quadratic factors and then use the Zero Product Property. Here's a general approach:

1. Set the Equation to Zero: Make sure the equation is in standard form (equal to $0$).
2. Factor out the Greatest Common Factor (GCF): This is always the first step. If every term shares a common factor, pull it out front. For example, in $x^3-4x=0$, the GCF is $x$, so you can write it as $x(x^2-4)=0$.
3. Look for Patterns: Can you factor what's left? Look for special cases like a difference of squares ($a^2-b^2=(a-b)(a+b)$) or sum/difference of cubes.
4. Factor by Grouping: If you have four terms, try factoring by grouping. This involves pairing up terms and factoring out the GCF from each pair.

What Are Common Mistakes When Solving Polynomial Equations?

Solving polynomial equations involves many steps, which means there are many places to make a small error. Being aware of these common pitfalls can help you double-check your work and improve your accuracy.

• Forgetting to Set the Equation to 0: The Zero Product Property only works if the product equals zero. If you have $(x-4)(x+1)=5$, you cannot assume $x-4=5$. You must first multiply it out, subtract 5 from both sides to set it to zero, and then re-factor.
• Sign Errors: A misplaced negative sign is one of the most frequent mistakes, especially when using the quadratic formula. Be extra careful when calculating $-b$ if $b$ is already negative, or when calculating $b^2-4ac$ if $c$ is negative.
• Dividing by a Variable: It can be tempting to simplify an equation like $x^2=5x$ by dividing both sides by $x$ to get $x=5$. However, this makes you lose a solution! The correct method is to set it to zero ($x^2-5x=0$), factor ($x(x-5)=0$), and find both roots: $x=0$ and $x=5$. Never divide by a variable unless you know it cannot be zero.
• Incomplete Factoring: After factoring once, always check if any of the remaining factors can be factored further. Forgetting to factor a difference of squares like $x^2-9$ is a common oversight.
• Square Root Errors: When solving an equation like $x^2=16$ by taking the square root, remember that there are two solutions: $x=4$ and $x=-4$. It's easy to forget the negative root.

Quick Summary: Key Concepts

Here is a quick reference guide to the most important ideas about polynomial equations:

• Polynomial Equation: An equation with variables raised to non-negative integer exponents, set equal to zero.
• Degree: The highest exponent in the equation. It determines the equation's name (Linear, Quadratic, Cubic) and its maximum number of solutions.
• Standard Form: Arranging terms in descending order of exponents on one side of the equation, with the other side being $0$. Example: $a{x}^2+bx+c=0$.
• Roots / Solutions: The values of the variable that make the equation a true statement.

Solving Strategy by Degree:

• Degree 1 (Linear): Isolate the variable using inverse operations.
• Degree 2 (Quadratic): Put in standard form. Solve by factoring if possible. If not, use the Quadratic Formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.
• Degree 3+ (Cubic, etc.): Put in standard form. Solve by factoring (look for GCF, use grouping, or identify special patterns) and then apply the Zero Product Property.