What Is A Polynomial? A Complete Guide
Polynomials serve as fundamental components in algebra. They play a pivotal role in various fields, including calculus, physics, and engineering.
But what is a polynomial, exactly? Learn about the different variables in a polynomial and how to solve them!
Understanding Mathematical Expressions: Polynomials
A polynomial is a mathematical expression that consists of variables raised to non-negative integer powers. These variables are combined through addition, subtraction, and multiplication.
Their structure, characterized by coefficients and exponents, enables the formulation of equations that model diverse phenomena. An understanding of polynomials is essential for advanced mathematical study and application.
Components
A polynomial can be as straightforward as a single block with a number on it, such as 5, which is actually a very simple polynomial by itself. It can also be more elaborate, such as 2x2 + 3x + 7.
The letters we use are called variables, and they can have powers or exponents, like the 2 in x2 which tells us the x block is used twice in multiplication. The numbers that appear in front of the variables, like the 2 in 2x2, are known as coefficients.
It’s important to note that in polynomial equations, the powers on the variables must be whole numbers, which means no fractions or decimals, and they can’t be negative either. So, a variable to the power of 3/2 or -1 would not be part of a polynomial.
Order
Polynomials follow a specific order, usually arranged from the largest power to the smallest. This order makes them easier to understand and work with.
For example, the polynomial 4x3 + x2 – 5x + 6 starts with the highest power of x, which is 3, and goes down to the smallest, which is the number 6 without any variable.
The Building Blocks: Terms of a Polynomial
A polynomial is built up of different parts called terms. Each term is a piece of the overall expression.
For instance, take the expression 4x3 + 2x – 1. It may look complicated, but it’s just three terms added and subtracted together.
- The first term is 4x3, which combines a number (4) and a variable (x) raised to the power of 3.
- Next, we have the second term, 2x. This one is a bit simpler, with the variable x only being multiplied by 2.
- And finally, there’s the third term, -1. This term doesn’t have a variable at all; it’s just a number, known as a constant.
Degree of a Polynomial
The degree of the polynomial is the highest exponent of the variable that you’ll find in the expression. The degree of the polynomial with more than one variable is equal to the sum of the exponents of the variables in it.
For example, in the expression x2 + 7x + 4, look for the largest exponent; here, it’s 2. This number tells us the degree is 2 because the highest power of x is x2.
The degree of a polynomial allows us to take a look into the polynomial’s characteristics. It can tell us how many roots or solutions the polynomial can have. It also hints at the shape of the graph that the polynomial will make when plotted on a coordinate plane.
Types Of Degree Of A Polynomial
Polynomials can be categorized based on their degree, which refers to the highest power of the variable in the expression.
There are mainly three types of polynomials:
- Linear polynomials, which have a degree of one;
- Cubic polynomials, characterized by a degree of three;
- Zero polynomials, where the polynomial evaluates to zero for all values of the variable.
The properties of polynomials enable various operations, including factoring polynomials and grouping like terms.
Different Towers: Types of Polynomials
There a multiple types of polynomials, differing in the number of terms
- Monomials: A monomial is the simplest type of polynomial. It has just a single term. An example of a monomial would be 7x4. Here, there’s only one ‘piece’ to look at, which is 7 times x raised to the fourth power.
- Binomials: Next up, we have binomials. They have exactly two terms. For instance, take x + 6. This binomial has two separate parts, x and 6. They’re distinct but part of the same structure.
- Trinomials: Moving on to a bit more complexity, we encounter trinomials. A trinomial such as x2 + 2x + 1, consists of three terms.
- When there are more four terms or more, we just call it a polynomial. Each term adds to the height of the mathematical structure.
Standard Form of Polynomials
There’s a specific way to write polynomials that makes them easier to understand. This method is called the standard form.
The standard form of a polynomial requires that you write the terms in order of their descending power of the variable, starting with the highest degree and going down to the lowest.
For example, x2 + x + 1 is in standard form because the terms go from the squared term (highest degree) down to the constant term (lowest degree).
Benefits Of Writing In Standard Form
Writing polynomials in standard form allows anyone reading the expression to quickly see the most ‘significant’ parts first—the terms with the highest powers of x.
Moreover, organizing polynomials also sets you up for success when you’re adding, subtracting, or comparing polynomials. You can easily match like terms (terms with the same power of x) and perform the necessary calculations without confusion.
An Example
For instance, if you have 3x2 + 4x + 5 and you want to add 2x2 – x + 3, having both polynomials in standard form makes it straightforward.
You simply add the coefficients (the numbers in front) of the like terms: (3 + 2)x2 for the x-squared terms, (4 – 1)x for the x terms, and 5 + 3 for the constants. Your answer, 5x2 + 3x + 8, is also in standard form, tidy and clear.
Mathematical Functions: Operations on Polynomials
Let’s explore how we can have fun with these math structures by performing different operations.
Adding Polynomials: Building Bigger Structures
Adding polynomials means adding the coefficients (the numbers in front) of terms that have the same variable raised to the same power.
For example, if you have 2x + 3 and 4x + 5, you simply add the ‘x’ terms and the constant terms together to get 6x + 8.
Subtracting Polynomials: Removing Blocks
Subtracting is just the opposite of adding. Instead of putting together, you’re taking away. With polynomials, you subtract the coefficients of like terms.
If we take 5x2 – 2x + 7 and want to subtract 3x2 + x – 4 from it, we end up with 2x2 – 3x + 11. Remember to switch the sign of each term in the second polynomial before you subtract.
Multiplying Polynomials: Creating Complex Designs
Multiplying polynomials can be compared to connecting blocks to make more intricate designs. You multiply each term in one polynomial by every term in the other polynomial.
For instance, if you multiply (x + 2) by (x – 3), you need to use the distributive property. Multiply each term in the first polynomial by each term in the second polynomial to get x2 – 3x + 2x – 6, which simplifies to x2 – x – 6.
Dividing Polynomials
There are different methods of performing division operations, such as long division or synthetic division, which work well with higher-degree polynomials. Let’s say you have 2x2 + 6x + 4 and you want to divide it by x + 2.
Using long division, you’d find that each ‘share’ is 2x + 2, with no leftovers. This means that every piece has been shared out evenly.
The Puzzle: Solving Polynomials
A polynomial is a math expression that can have constants and variables, and exponents arranged in a particular way. To solve it, you’re essentially looking for the value that will make the polynomial equal to zero.
Single Variable Polynomials
Now, if you have a polynomial with just one variable and it’s not raised to a high power, you can solve it by performing simple arithmetic operations For example, if you have x + 3 = 0, you can easily find that x = -3.
Multiple Variables
However, when the polynomial gets more complex, with variables raised to higher powers or multiple variables, solving becomes more challenging. In these cases, you might use different methods like factoring, using the quadratic formula, or even graphing.
For example, with a quadratic polynomial expression (which is a type of polynomial with a variable squared, like x^2), you can use the quadratic formula to find the variable’s values that make the equation zero.
Lastly, remember that some polynomials have more than one solution. These are known as the “roots” or “zeroes” of the polynomial.
Polynomials: The Language of Algebra
In conclusion, polynomials can be simple or complex, but they all follow certain rules. By learning about polynomials, you’re learning the language of algebra, which is a powerful tool in math. By understanding these terms, you’re on your way to mastering the basics of algebraic expressions!
