In mathematics, we use letters to perform calculations. This is called algebraic expression. Before you study the algebraic expression, you use only numbers in the calculation. However, after learning algebraic expressions, math is mainly done by using alphabets.

Algebraic expressions have rules. Therefore, you have to learn how to use algebraic expressions.

There are also several types of equations that are represented by symbols: monomials and polynomials. What is the difference between these formulae? If you don’t understand the rules of algebraic expressions, you can’t do the addition and subtraction of these equations.

So, we’ll explain the rules of algebraic expressions, the difference between monomial and polynomial formulas, and how to perform addition and subtraction of these equations.

## In an Algebraic Expression, Turning Unknown Numbers into Symbols

What kind of calculation is the algebraic expression that we learn in math? In an algebraic expression, we use letters of the alphabet to represent equations as follows.

• $10+a$
• $2x+y$
• $2a^2b$

When we look at equations that contain symbols, the content seems to be difficult.

However, we have been studying algebraic expressions in elementary school. For example, what is included in the following $□$?

• $10-□=4$

You must have solved a problem like this before. The number that goes in □ is 6. For numbers we don’t know, we make an equation by using □. Instead, in the algebraic expressions, we use letters to represent numbers we don’t know.

For example, in the previous question, we can replace $□$ with $x$ and write the following.

• $10-x=4$

This is an algebraic expression. We use letters of the alphabet, such as $x, y, a, b$, etc., to represent numbers we don’t know, to create an equation.

-Algebraic Expressions Are Useful in Calculations

Why do we use algebraic expressions? It is because they are useful in math calculations. Mathematics is used frequently in our daily life. In most of those cases, we already know the answer, but we don’t know what to do with the process on the way.

For example, here’s an example.

• How many kilometers per day do you need to walk 1,000 km in 365 days (one year)?

We already know the answer to walking 1,000km. But we don’t know the process of how many kilometers a day should we walk. So, we replace the numbers we don’t know with symbols such as $x$, and calculate them.

Even if we don’t know the numbers on the way, we can construct the equation and calculate it. This is the reason why we use a lot of algebraic expressions in mathematics.

### Understand the Rules of Algebraic Expressions

What are the rules in algebraic expressions? Be sure to follow the following rules in algebraic expressions.

• Omit the × in multiplication.
• Write the numbers before the letters and put them in alphabetical order.
• Multiplication of the same letters use exponents.
• Leave out 1 and -1.
• Don’t use division, use multiplication of fractions

As for the rules, the content is not difficult to follow. Just make sure you understand them.

-Omit × in Multiplication.

Before learning the algebraic expression, we always use the symbol × when multiplying. However, in algebraic expressions, the × is omitted when multiplying. Therefore, you get the following.

It is written as $2b$, not $2×b$. The other is written as $ab$, not $a×b$. In the algebraic expression, the × is omitted.

-Write the Numbers in Before the Letters and Put Them in Alphabetical Order

In the algebraic expression, what order should we write them in? The rule is that the numbers must always come before the letters of the alphabet. Also, if there are several letters, they should be written in alphabetical order.

For example, the following multiplication is expressed in the following algebraic expressions.

• $a×4×b=4ab$

The number 4 is written first. The letters are written in alphabetical order, so they are written as $4ab$.

-Multiplication of the Same Letters Using Exponents

In algebraic expressions, we frequently multiply the same letter. For example, $a×a$. However, we never write $aa$ in an algebraic expression. In multiplication of the same letter, we use exponents. For example, we have the following.

• $a×a=a^2$
• $3×y×y×y=3y^3$

-Leave Out 1 and -1

If we write in numbers at the beginning, we think that we should write them as follows.

• $1×a=1a$
• $-1×b=-1b$

However, we have to leave out the 1 and -1. 1 can be multiplied infinitely, so we can do “$1×1×1×a$” and so on. To prevent this, we leave out the numbers 1 and -1. For -1, we write the $-$, as in $-b$, and so on. Therefore, we get the following.

• $1×a=a$
• $-1×b=-b$

-Don’t Use Division, Use Multiplication of Fractions

We don’t use division in algebraic expressions. Division is limited in its use, and few people use division in mathematics. In fact, no one uses division in junior high school or high school math.

Division can be corrected to multiplication of fractions. Don’t use division in algebraic expressions, just multiplication equations.

### Learn the Algebraic Expressions for Fractions and Reciprocal Numbers

One of the most difficult concepts to understand when understanding algebraic expressions is to convert division into multiplication of fractions. So, we’ll explain how to change the division of algebraic expressions into fractions.

Divisions can be converted to multiplication of fractions by using the reciprocal, as follows.

The same is true for algebraic expressions. The division of algebraic expressions can be corrected to multiplication of fractions by taking the reciprocal, as shown below.

Note that in the algebraic expression, you can write the alphabet in the numerator or next to the fraction. Therefore, it is as follows.

Both notations are correct. You can use either, but you should be able to write both.

As a reminder, if there are letters in the denominator, the alphabet must be in the denominator. Do not write the alphabet next to the fraction. Therefore, it is as follows.

As mentioned above, if there are alphabets in the numerator, you can write them in the numerator or next to the fraction. However, if there are alphabets in the denominator, be sure to write them in the denominator.

## The Difference Between Monomials and Polynomials in Algebraic Expressions

After understanding the rules about these algebraic expressions, we need to understand the meaning of the words. When you study algebraic expressions, the words you must learn are monomials and polynomials. Both monomials and polynomials are equations that make use of letters and numbers.

The differences between them are as follows.

• Monomials: equations that only involve multiplication
• Polynomials: equations that mix addition and subtraction

You can understand that a monomial is, in essence, a single letter equation.

We explained that in an algebraic expression, you omit the × when you multiply. As a result, in an algebraic expression with only multiplication, the numbers and symbols will line up. In this case, the mass of numbers and letters is a monomial.

However, in mathematics, the equation does not necessarily include only multiplication. While division can be corrected to multiplication, addition and subtraction cannot be corrected to multiplication.

So if there are additions and subtractions in an equation, there are several monomials. When an equation contains addition and subtraction, and there are multiple monomials, it is called a polynomial.

The difference between monomials and polynomials is determined by whether addition or subtraction is included in the equation. It is a polynomial that has more than one monomial in the equation, using plus or minus.

### Coefficients, Terms and Degrees of Monomials and Polynomials

For reference, when we learn monomials and polynomials in algebraic expressions, the terms coefficients, terms, and degrees are mentioned. They are as follows, respectively.

• Coefficient: a number in a monomial expression
• Term: a monomial including numbers and letters
• Degree: numbers of letters, such as $x$ and $a$

You don’t need to understand and remember the definitions of these words. However, these words are always found in textbooks that explain monomials and polynomials, so you should understand roughly what they mean.

As for the coefficients, they refer to the numbers in the monomial. For example, it is as follows.

Understand that the numbers in front of letters, including plus and minus, are coefficients.

On the other hand, each monomial is called a term. In the previous equation, the following are the terms.

It is easy to distinguish between coefficients and terms. However, coefficients and terms are not important words in mathematics. A more important word is the degree.

The degree refers to the number of letters (alphabets) contained in a monomial.

• One letter: first degree (linear)
• Two letters: second degree (quadratic)
• Three letters: third degree (cubic)
• Four letters: fourth degree (quartic)

Note that a polynomial contains multiple monomials of different degree. Among these monomials, the highest number of letters is used as the degree.

It is important to understand the definition of degree in polynomial. The highest number of letters of monomials in a polynomial is the degree of the polynomial. For example, the following equation contains several monomials.

In this equation, the largest degree is 2. Therefore, this polynomial is quadratic.

In mathematics, we learn formulas such as linear and quadratic equations. How do we distinguish whether it is a linear or a quadratic equation? In this regard, we can distinguish between them depending on how many degrees the polynomial has.

## Addition and Subtraction of Polynomials: Combining Like Terms (Similar Term)

After learning about monomials and polynomials in algebraic expressions, the next step we need to understand is addition and subtraction. We’ve already learned about addition and subtraction. But how can we do addition and subtraction for algebraic expressions that contain alphabets?

The concept of these algebraic expressions is the same as for numbers, even if the equation contains letters like $x$ and $y$. For example, suppose we have the following equation.

• $3x+1$

You can put any number in $x$. So the answer changes as follows.

It is important that even in algebraic expressions, the answer can be a specific number.

So in an algebraic expression, if the letters are exactly the same, they can be put together by addition or subtraction. In polynomials, terms with the same letters are called like terms (or similar terms). We can distinguish like terms as follows,

If the letters are exactly the same, they are like terms and can be grouped together. In other words, you can add and subtract similar terms to each other.

Grouping similar terms by adding and subtracting is called combining like terms. In the case of combining like terms, it is as follows.

After lining up the like terms, add or subtract the coefficients from each other. For example, the coefficient on $x^2$ is 1. And the coefficient on $-3x^2$ is -3. Therefore, we get the calculation shown in the above figure.

Once again, it is written as follows

• $x^2-3x^2$
• $=(1-3)x^2$
• $=-2x^2$

In addition and subtraction of polynomials, combine like terms. To do this, you need to arrange similar terms. A similar term is a monomial whose letters are exactly the same but whose numbers (coefficients) are different.

If the degrees are different, they are not like terms. For example, the following are not similar terms.

Only monomials whose letters are exactly the same, including the degree, are like terms. For example, the following monomials are not similar terms.

• $2x^2$
• $4xy$
• $3a^2$
• $-3x$
• $5y$

Why are these not the like terms? It’s because the letters and degrees are different. Understand that you can only combine like terms when the letters are exactly the same. You can add or subtract like terms because they have the same properties.

### When Subtracting Polynomials, Be Careful of the Negative

Note that in addition and subtraction of algebraic expressions, we must be especially careful in subtraction. When subtracting polynomials with parentheses, many people make a miscalculation. To be more specific, if there is a minus sign before the parentheses, a lot of calculation errors occur.

If there is a minus sign before the parentheses, the sign changes when the parentheses are removed. For example, it is as follows.

In other words, all the signs in a parenthesis change. Why does having a negative in front of a parenthesis change all the signs in the parenthesis? The reason is that equations in parenthesis is considered to be a number.

In mathematics, there is a rule that parentheses must be calculated first. The reason for this is that even if there are equations in parentheses, they are considered to be numbers. Because it is a single number, it must be calculated first.

For example, if there is the equation $-(1+3)$, how do you calculate it? We can calculate it as follows.

In this equation, we do $1+3$ first because of the parenthesis. Then, by multiplying by the minus sign, the answer is -4.

On the other hand, if we remove the parenthesis first, how can we calculate? In this case, both numbers are multiplied by -1. This means that the all signs in the parenthesis must be changed. As a result, the answer is the same as in the previous example, as shown below.

As for the equation in parenthesis, it must be considered the same number. This is why we must change all signs when removing parentheses if there is a minus sign before the parentheses. Because they are considered the same, we must not change the sign for only one of the numbers in the equation when we remove the parentheses.

Therefore, when we combine similar terms, we can do the following.

For addition of polynomials, we can solve the problem without any trouble. On the other hand, many people make mistakes in polynomial subtraction. There are certain parts of polynomial subtraction that make calculation errors in. When you remove the parentheses in subtraction, a lot of miscalculations occur.

Understand where you are most likely to make a calculation error. Also, learn how to answer the question correctly. Then you will be able to avoid calculation errors in addition and subtraction of polynomials.

## Exercises: Addition and Subtraction of Monomials and Polynomials

Q1. Do the following calculation.

1. $2a÷(-5)$
2. $3b÷2x÷y$

In calculating a monomial, you have to correct the division into multiplication. So let’s use the reciprocal and convert them all to multiplication of fractions. Then we get the following.

(a)

$2a÷(-5)$

$=2a×\left(-\displaystyle\frac{1}{5}\right)$

$=-\displaystyle\frac{2a}{5} \left(or-\displaystyle\frac{2}{5}a\right)$

(b)

$3b÷2x÷y$

$=3b×\displaystyle\frac{1}{2x}×\displaystyle\frac{1}{y}$

$=\displaystyle\frac{3b}{2xy} \left(or\displaystyle\frac{3}{2xy}b\right)$

Q2. Do the following calculation.

• $\displaystyle\frac{1}{2}x-\displaystyle\frac{1}{3}x$

You can add or multiply the like terms in algebraic expressions. Of course, even fractions and decimals can be added or subtracted between like terms.

$\displaystyle\frac{1}{2}x-\displaystyle\frac{1}{3}x$

$=\displaystyle\frac{3}{6}x-\displaystyle\frac{2}{6}x$

$=\displaystyle\frac{1}{6}x$

Q3. Do the following calculation.

1. $x^2+3-3x-7x^2+5x-10$
2. $(-2x+4y)+(9x-8y)$
3. $(x+3y)-(3x-6y)+4x$

In polynomials, try to combine like terms with each other. This way, you can add and subtract.

If the sign before the parenthesis is positive, there is nothing to pay attention to. On the other hand, if the sign before the parentheses is negative, you will often make a mistake when removing the parentheses. When removing parentheses, it is necessary to change the sign.

(a)

$\textcolor{red}{x^2}+3-\textcolor{blue}{3x}\textcolor{red}{-7x^2}+\textcolor{blue}{5x}-10$

$=\textcolor{red}{x^2- 7x^2}\textcolor{blue}{-3x+5x}+3-10$

$=\textcolor{red}{(1-7)}x^2+\textcolor{blue}{(-3+5)}x+(3-10)$

$=-6x^2+2x-7$

(b)

$(\textcolor{red}{-2x}+\textcolor{blue}{4y)}+(\textcolor{red}{9x}\textcolor{blue}{-8y})$

$=\textcolor{red}{-2x+9x}+\textcolor{blue}{4y-8y}$

$=\textcolor{red}{(-2+9)}x+\textcolor{blue}{(4-8)}y$

$=7x-4y$

(c)

$(x+3y)\textcolor{red}{-(3x-6y)}+4x$

$=x+3y\textcolor{red}{-3x+6y}+4x$

$=x-3x+4x+3y+6y$

$=(1-3+4)x+(3+6)y$

$=2x+9y$

## Learn the Definition and How to Calculate Algebraic Expressions

After studying algebraic expressions in mathematics, almost all calculation exercises are algebraic expressions. Therefore, if you don’t understand the rules of algebraic expressions, you won’t be able to solve math calculation problems. So be sure to learn rules of the algebraic expressions.

The rules make it possible for us to do the correct calculation. And don’t just remember the rules; learn the reason why.

Note that when adding and subtracting polynomials, it’s important to combine similar terms. Especially if there is a minus sign before the parentheses, calculation errors are more likely to occur. Solve the problem by making sure that the signs for the plus and minus are correct.

A very important study in mathematics is algebraic expressions. Since algebraic expressions are found in every math question, make sure you learn the basics, including the definition and even the reason for the algebraic expressions.