Algebraic expressions are math equations that use the alphabet. In junior high school and high school math, we need to understand how to use algebraic expressions because they are almost always used in math.

An equation that consists of several letter expressions is called polynomials. Calculation of polynomials frequently involves multiplication and division. So we must understand not only addition and subtraction, but also how to multiply and divide polynomials.

In order to understand this, we need to learn the distributive property (or distributive law). Understanding the distributive property is the key to being able to multiply and divide polynomials.

Also, once you learn multiplication and division of polynomials, you will be able to perform substitution calculations. We will explain how to do multiplication and division of polynomials and substitution.

## Expansion of the Equation: Remove the Parentheses with the Distributive Law

There are several laws in mathematics. One of the most important laws is the distributive property. If you study mathematics, we all use the distributive law.

Let’s understand that the distributive property is a rule that allows you to remove parentheses. Specifically, it is as follows.

This is one of the most basic formulas and is frequently used in polynomial calculations. By using the distributive property, we can remove the parentheses and convert it into addition or subtraction equations.

Removing the parentheses is called expanding (or developing) in mathematics. When a math problem tells you to expand, it means to remove the parentheses by using the distributive property.

### The Reasons Why the Distributive Property Established, Think in terms of Area

Why does the distributive property hold? The reason for this is easy to understand if you think of it in terms of area addition. The distributive property is taught in elementary school or junior high school mathematics. The reason why we learn it in elementary school is that we can explain the reason why the distributive law is valid by using a simple calculation of area.

When we find the area of a rectangle, we use the following formula.

• Vertical x Horizontal = Area

Therefore, the area of the two rectangles below can be calculated by the following formula.

However, when calculating the area, it’s okay to add the horizontal length first. That is, you can add $b + c$ first.

After adding $(b + c)$ to get the horizontal length, we multiply it by $a$, the vertical length. This method also gives us the total area of the two rectangles.

In other words, we see that the following formula holds.

• $a(b+c)=ab+ac$

In junior high school and above, all people in mathematics use the distributive property in the development of equations. The reason for the distributive law can be explained by elementary school math.

## Use the Distributive Property in Algebraic Multiplication and Division

When multiplying with alphabets, you need to understand the distributive property to calculate them. In literal formulas, we frequently use the formula we just discussed.

In algebraic expressions, there are many situations in which you have to develop the equation as shown below. How do we solve such equations?

• $2(3x-4y)$
• $3a(a+3)$

The distributive property holds not only for numbers, but also for letters. So when we expand the equation, we can multiply all the equations in parentheses like below.

If there are equations with plus or minus in parentheses, multiply each one by all of them.

### Multiply All Numbers in Parentheses by Minus or Fractions

However, when calculating using the distributive law, calculation errors often occur. In multiplication and division of polynomials, miscalculation is more likely to occur in the following cases.

• Negative multiplication
• Multiplication of fractions

In the distributive property, all the equations in the parentheses are multiplied. So, if you want to multiply minus sign by the distributive property, you must multiply all the equations in the parentheses by the minus sign. This means that all the signs will change.

When expanding an equation, it is wrong to change the sign of only one of the equations in parentheses. For example, the following calculation is wrong.

• $−2(3+1)=-6\textcolor{red}{+}2$

The same can be said for fractions. In multiplying fractions, if a negative sign is in front of a fraction, you must do the following.

• $-\displaystyle\frac{2x-3}{4}=\displaystyle\frac{-2x+3}{4}$

$-\displaystyle\frac{2x-3}{4}$ has the same meaning as $\displaystyle\frac{-(2x-3)}{4}$. So when you remove the parentheses, make sure they are in the correct order.

### Divisions Are Corrected to Multiplication of Fractions and Perform Reduction

Note that in math calculations, we don’t use division in all equations after we learn the algebraic expressions. By using the reciprocal, we can always convert them into multiplication of fractions. Just be sure to pay attention to the numerator and denominator when you take the reciprocal.

For example, $\displaystyle\frac{2}{3}x=\displaystyle\frac{2x}{3}$. Therefore, the reciprocal of $\displaystyle\frac{2}{3}x$ is $\displaystyle\frac{3}{2x}$. When making the reciprocal of $\displaystyle\frac{2}{3}x$, it must not be $\displaystyle\frac{3}{2}x$.

So, for example, we have the following calculation.

There are several points of miscalculation in the multiplication (or division) with algebraic expressions. You need to understand which parts of the equation are most likely to be miscalculated and make sure that the signs and reciprocals are correct.

## Putting Numbers into Algebraic Expressions by Substitution

Why do we need to learn how to calculate polynomials? Because it can be corrected into simple calculations.

The reason why we need to learn math is because it is used frequently in our daily lives. Mathematics is used in all kinds of situations such as bookkeeping, programming, and marketing, as well as science, to perform calculations. So, we create letter equations and put in numbers to get the answer.

For example, we have the following equation.

• $3x+1$

The answer to this equation depends on what numbers you apply to it, as follows.

As you can see, fitting a number to algebraic expressions is called substitution. Substitution allows you to get the answer you’re looking for.

### Calculating Polynomials to Simplify Equations

By using substitutions, we can get a specific answer. For example, if $x=3$, then by substitution we get the following.

• $2x-2=2×3-2=4$
• $x^2=3×3=9$

If the algebraic expressions are simple, the problem can be solved without any difficulty. On the other hand, if $x=-3,y=2$, what does the answer look like if we substitute in the following equation?

• $-3(3x-5y)+4(8x-4y)$

By substituting to this equation, it is no problem to calculate $-3[3×(-3)-5×2]+4[8×(-3)×-4×2]$. However, the calculation is quite complicated.

So in the algebraic expressions, we calculate the algebraic expressions first. Therefore, we have the following.

Calculating the algebraic expression yields $23x-y$. After simplifying the algebraic expression by addition, subtraction, multiplication, and division, it is easy to calculate the substitution. If $x=-3,y=2$, then the result is as follows.

$23x-y$.

$=23×-3 -2$

$=-69-2=-71$

When you substitute, calculate the algebraic expressions first. Then, substitute numbers into the alphabet to get the answer.

## Exercises: Calculating and Substituting Polynomials

Q1. Do the following calculation.

1. $3(4x+3)-4(2x-1)$
2. $\displaystyle\frac{2x+3}{2}-\displaystyle\frac{x+4}{3}$

In polynomial multiplication, you have to be careful when removing the parentheses. If there is a minus sign before the parentheses, then all the signs change when the parentheses are removed.

The same is true for fractions. If a minus sign is present in front of a fraction, the sign will change according to the distributive property. So we get the following calculation.

(a)

$3(4x+3)-4(2x-1)$

$=12x+9-8x+4$

$=4x+13$

(b)

Q2. if $x=3$ and $y=-2$, do the following calculation.

1. $-3(2x+2y)+2(4x-5y)$
2. $(2x^2+4xy)÷\displaystyle\frac{2}{3}x$

In the calculation of substitutions, calculate the algebraic expressions first. If you do so, it will be easier to calculate when you substitute numbers.

(a)

$-3(2x+2y)+2(4x-5y)$

$=-6x-6y+8x-10y$

$=2x-16y$

• Substitute $x=3,y=-2$.

$2x-16y$

$=2×3-16×(-2)$

$=6+32=38$

(b)

$(2x^2+4xy)÷\displaystyle\frac{2}{3}x$ : Move $x$ to the numerator

$=(2x^2+4xy)÷\displaystyle\frac{2x}{3}$ : Convert to multiplication

$=(2x^2+4xy)×\displaystyle\frac{3}{2x}$ : Distributive law

$=2x^2×\displaystyle\frac{3}{2x}+4xy×\displaystyle\frac{3}{2x}$ :Reduce

$=3x+6y$

• Substitute $x=3,y=-2$.

$3x+6y$

$=3×3+6×(-2)$

$=9-12=-3$

## Learn How to Do Distributive Property and Polynomial Calculations

Mathematics is almost always calculated in algebraic expressions. Therefore, you need to understand how to calculate polynomials. After learning addition and subtraction, the next most important thing to learn is multiplication and division of polynomials.

In order to multiply (or divide) a polynomial, you must learn the distributive property. We have explained why the distributive law is established in an easy to understand way. Be sure to understand the principles of the distributive property.

In addition, there are some points that many people make mistakes in calculating the algebraic expressions using the distributive property. It is the following part.

• The minus exists before the parenthesis.
• The minus exists before the fraction.
• Use reciprocals to convert division into multiplication of fractions.

If there is a minus sign before a parenthesis or a fraction, pay attention to whether the sign is positive or negative when expanding the equation by using the distributive property. Also, when using the reciprocal, make sure that the letter is in the numerator or denominator. If you pay attention to these things, you will be able to avoid miscalculation.