Polynomials

What Is a Polynomial?

A polynomial is an algebraic expression made up of one or more terms, where each term consists of a coefficient multiplied by a variable raised to a non-negative integer power. That might sound complicated, but the idea is simpler than it seems. Let's break down the key ingredients:

• Variables: These are the letters in the expression, like $x$ or $y$.
• Coefficients: These are the numbers that multiply the variables. In the term $7x^2$, the coefficient is $7$.
• Exponents: These are the small numbers written above and to the right of the variables. In $7x^2$, the exponent is $2$. A crucial rule for polynomials is that exponents must be non-negative whole numbers ($0,1,2,3,...$).
• Terms: Terms are the individual parts of the polynomial that are separated by addition ($+$) or subtraction ($-$) signs.

For example, the expression $5x^3-2x^2+x-9$ is a polynomial. It has four terms: $5x^3$, $-2x^2$, $x$, and $-9$.

However, not every algebraic expression is a polynomial. An expression is not a polynomial if it contains:

• Negative exponents: For example, $3x^{-2}+4$ is not a polynomial because of the $-2$ exponent.
• Fractional exponents: For example, $x^{1/2}-7$ is not a polynomial. This is another way of writing $\sqrt{x}-7$.
• Variables in the denominator: For example, $\frac{6}{x}+5$ is not a polynomial, because this is the same as $6x^{-1}+5$.

Polynomials are the foundation of algebra, and understanding their structure is the first step toward mastering them.

The Anatomy of a Polynomial: Terms, Degrees, and More

To work with polynomials effectively, you need to know the specific vocabulary used to describe their parts. Let's dissect a typical polynomial, $-4x^5+8x^3-x+12$, and identify each component.

• Terms: As we learned, terms are the parts separated by $+$ or $-$. The terms here are $-4x^5$, $8x^3$, $-x$, and $12$.
• Degree of a Term: The degree of a single term is its exponent. The degree of $-4x^5$ is $5$, and the degree of $8x^3$ is $3$. The term $-x$ can be written as $-1x^1$, so its degree is $1$.
• Constant Term: A term without a variable, like $12$, is the constant term. You can think of it as $12x^0$, so its degree is $0$.
• Degree of the Polynomial: This is the highest degree of any single term in the polynomial. For $-4x^5+8x^3-x+12$, the highest exponent is $5$, so the degree of the entire polynomial is $5$.
• Standard Form: A polynomial is in standard form when its terms are written in descending order of degree. Our example polynomial is already in standard form.
• Leading Term: When a polynomial is in standard form, the very first term is the leading term. Here, it is $-4x^5$.
• Leading Coefficient: This is the coefficient of the leading term. In our example, the leading coefficient is $-4$.

Here is a table to summarize the parts of $-4x^5+8x^3-x+12$:

Understanding this terminology is essential because it provides a precise way to describe and compare different polynomials.

How Do You Classify Polynomials?

Polynomials can be classified, or named, in two primary ways: by the number of terms they contain and by their degree. This naming system gives us a quick and descriptive snapshot of the polynomial we're dealing with.

Classification by Number of Terms

This is the most straightforward classification. It's simply a count of how many terms are in the expression after all like terms have been combined.

• Monomial: A polynomial with just one term. Examples: $7x$, $-5$, $12y^4$.
• Binomial: A polynomial with two terms. Examples: $x+4$, $3x^2-9$.
• Trinomial: A polynomial with three terms. Examples: $x^2+6x+5$, $4a^3-2a+1$.

Any polynomial with four or more terms is simply called a "polynomial with $n$ terms," for example, a "polynomial with 4 terms."

Classification by Degree

This classification is based on the degree of the polynomial (the highest exponent of the variable).

• Degree 0: A non-zero constant. Example: $8$. This is called a Constant.
• Degree 1: The highest exponent is $1$. Example: $5x-2$. This is called a Linear polynomial.
• Degree 2: The highest exponent is $2$. Example: $3x^2-4x+1$. This is called a Quadratic polynomial.
• Degree 3: The highest exponent is $3$. Example: $x^3+2x^2$. This is called a Cubic polynomial.
• Degree 4: The highest exponent is $4$. Example: $6y^4-y$. This is called a Quartic polynomial.

Often, we use both classifications to describe a polynomial. For instance, $x^2+6x+5$ is a quadratic trinomial, and $5x-2$ is a linear binomial.

How to Add and Subtract Polynomials

Adding and subtracting polynomials is all about identifying and combining like terms. Like terms are terms that have the exact same variable part, meaning the same variable(s) raised to the same power(s). For example, $7x^2$ and $-3x^2$ are like terms, but $7x^2$ and $7x$ are not.

Adding Polynomials

To add polynomials, you simply remove the parentheses and combine the like terms by adding their coefficients. You can arrange the problem horizontally or vertically.

Subtracting Polynomials

Subtraction is very similar, but with one critical extra step. When you subtract a polynomial, you must first distribute the negative sign to every single term inside the second set of parentheses. This changes the sign of each term. A common mistake is to only change the sign of the first term.

How to Multiply Polynomials

Multiplying polynomials involves using the distributive property repeatedly. The fundamental rule is that every term in the first polynomial must be multiplied by every term in the second polynomial. When you multiply terms, you multiply their coefficients and add their exponents (e.g., $(2x^2)(3x^4)=6x^{2+4}=6x^6$).

Multiplying a Monomial by a Polynomial

This is the simplest case. You just distribute the single term (the monomial) to each term inside the other polynomial.

For example, to multiply $4x^2(3x^2-5x+2)$:

Multiplying a Binomial by a Binomial (The FOIL Method)

When you multiply two binomials, a helpful acronym to remember the process is FOIL. This ensures you multiply all four pairs of terms.

Multiplying Any Two Polynomials

The FOIL method is just a special case of the distributive property. For larger polynomials, like a binomial times a trinomial, you can't use FOIL. Instead, you must systematically multiply each term from the first polynomial by each term in the second one.

What Are the Special Products of Binomials?

Certain binomial products appear so frequently in algebra that it's worth memorizing their patterns. Recognizing these special products allows you to find the answer quickly without going through all the steps of FOIL, and it is crucial for the reverse process, called factoring.

Perfect Square Trinomial (Sum)

This occurs when you square a binomial that involves addition.

Notice the middle term is twice the product of $a$ and $b$. For example, let's expand $(x+6{)}^2$. Here, $a=x$ and $b=6$.

Perfect Square Trinomial (Difference)

This occurs when you square a binomial that involves subtraction.

The only difference is that the middle term is negative. For example, let's expand $(2y-3{)}^2$. Here, $a=2y$ and $b=3$.

Difference of Squares

This pattern arises when you multiply two binomials that are identical except for the sign between their terms (these are called conjugates).

When you multiply conjugates, the middle terms (Outer and Inner from FOIL) always cancel out, leaving just the square of the first term minus the square of the second term. For example, let's multiply $(3x+5)(3x-5)$. Here, $a=3x$ and $b=5$.

Common Mistakes to Avoid with Polynomials

Working with polynomials is systematic, but there are a few common pitfalls that can trip up students. Being aware of these mistakes is the best way to avoid making them.

• Forgetting to Distribute the Negative: When subtracting polynomials, like in $(A)-(B)$, it is crucial to distribute the negative sign to every term in polynomial $(B)$, not just the first one.
• Combining Unlike Terms: You can only add or subtract terms that have the exact same variable and exponent. You cannot combine $5x^2$ and $3x$ into $8x^3$. They are not like terms and must be kept separate.
• Incorrect Exponent Rules: When multiplying terms, remember to add the exponents ($x^2\cdot x^3=x^{2+3}=x^5$). Do not multiply them ($x^2\cdot x^3\ne x^6$). When adding terms, the exponents do not change at all ($x^2+x^2=2x^2$).
• The Fake Shortcut for Squaring a Binomial: A very common error is to think that $(a+b{)}^2$ is equal to $a^2+b^2$. This is incorrect! You are missing the middle term. As we saw in special products, the correct expansion is $(a+b{)}^2=a^2+2ab+b^2$.
• Sign Errors with FOIL: Pay close attention to positive and negative signs when multiplying, especially for the Inner and Last terms. For example, in $(x-2)(x-7)$, the last term is $(-2)(-7)=+14$, not $-14$.

Polynomials: Quick Reference Guide

This section provides a quick summary of the key concepts for you to reference.

Key Definitions

• Polynomial: An expression with one or more terms, where variables have non-negative integer exponents.
• Standard Form: Terms are ordered from highest degree to lowest degree.
• Degree: The highest exponent in the polynomial.
• Like Terms: Terms with the identical variable part (e.g., $3x^2$ and $-5x^2$).

Classification Summary

Core Operations

1. Addition: Remove parentheses and combine like terms.
2. Subtraction: Distribute the negative sign to the second polynomial, then combine like terms.
3. Multiplication: Multiply every term in the first polynomial by every term in the second. Remember to add exponents when multiplying variables.