【8th Grade Math】Chapter 1: Algebraic Expressions
Hello everyone! Welcome to 8th-grade math.
Do you remember the "algebraic expressions" you learned in 7th grade? In this chapter, we are going to power up that knowledge and learn how to solve even more complex equations with ease.
"Algebraic calculations" are the super-important foundation for all the math you will learn from now on.
At first, you might feel like it looks difficult with all those variables, but the rules are actually very simple. It's just like solving a puzzle—let's enjoy mastering it together!
1. Monomials and Polynomials
First, let's organize our mathematical terminology. Once you understand what these words mean, word problems will become much easier to read.
Monomials and Polynomials
- Monomial: An expression consisting only of numbers and variables multiplied together. (e.g., \( 3x \), \( -5ab^2 \), \( y \))
- Polynomial: An expression represented as a sum (addition) of monomials. (e.g., \( 2x + 3y \), \( a^2 - 4a + 5 \))
Each individual monomial within a polynomial is called a "term."
For example, in the expression \( 2x - 5y \), the terms are \( 2x \) and \( -5y \). Remember that the minus sign stays with the term!
Degree
In a monomial, the number of variables multiplied together is called the "degree."
Example: \( 3x^2 \) has two \( x \)'s, so the degree is "2". \( 5abc \) has three variables, so the degree is "3".
For a polynomial, the highest degree among its terms is considered the degree of the entire polynomial.
Example: In \( x^2 + 3x + 1 \), the highest degree is "2" (from \( x^2 \)), so it is a "quadratic expression" (degree 2).
【Key Point】
When there is a number in front of a variable, that number is called the "coefficient."
The coefficient of \( -4x^2 \) is \( -4 \).
◎ Summary of this section:
・If the expression is "one chunk," it's a monomial; if it's connected by addition/subtraction, it's a polynomial.
・The degree is determined by "how many variables are being multiplied!"
2. Addition and Subtraction of Polynomials
Here, we will learn how to add and subtract expressions. The trick is to find the "like terms."
Combining Like Terms
Terms that have exactly the same variable parts are called "like terms."
Just like adding apples to apples, you can combine these into one.
\( 3a + 5b + 2a - b \)
\( = (3 + 2)a + (5 - 1)b \)
\( = 5a + 4b \)
(* Since \( a \) and \( b \) are different types, you cannot calculate them any further!)
Calculations with Parentheses (Watch out for subtraction!)
When subtracting polynomials, the ironclad rule is to change the sign of every term in the expression being subtracted.
\( (5x - 3y) - (2x - 8y) \)
\( = 5x - 3y - 2x + 8y \) ← Look here! \(-8y\) becomes \(+8y\).
\( = 3x + 5y \)
【Common Mistake】
A very frequent error is applying the minus sign in front of the parentheses only to the first term!
Think of it as "distributing the minus to everyone inside the parentheses."
◎ Summary of this section:
・You can only perform calculations on terms that have the same variables and degree (like terms)!
・When subtracting, flip all the signs inside the following parentheses!
3. Multiplication and Division of Monomials
Now, let's look at multiplying and dividing variables.
Multiplication
Multiply numbers with numbers, and variables with variables.
\( 4a \times 3b = 12ab \)
\( 2x \times 5x = 10x^2 \) (Since there are two \( x \)'s, it becomes \( x^2 \))
Division
The most reliable way to divide, with the fewest mistakes, is to "write it in fraction form."
\( 8ab \div 2a \)
\( = \frac{8ab}{2a} \)
Now, simplify the fraction. The numbers: \( 8 \div 2 = 4 \); the variables: \( a \) cancels out, leaving only \( b \).
Answer: \( 4b \)
【Fun Fact: Rules of Exponents】
Just like \( a^2 \times a^3 = a^5 \), when you multiply, you add the numbers at the top right (exponents). Conversely, for division, you subtract them!
◎ Summary of this section:
・For multiplication, multiply everything together!
・For division, turn it into a fraction and "cancel out" (simplify) the same variables from the top and bottom!
4. Applications of Algebraic Calculations (Explanation/Proof)
This is the section where we use algebra to explain mathematical mysteries like, "Why does it always turn out like this?" It might feel a bit difficult at first, but once you learn the patterns, you'll be fine!
Common Ways to Represent Variables
- Even number: \( 2n \)
- Odd number: \( 2n + 1 \) (or \( 2n - 1 \))
- Three consecutive integers: \( n, n+1, n+2 \)
(* \( n \) is an integer)
Tips for Writing Explanations
For example, to explain that "the sum of two odd numbers is an even number":
1. Declare the variables you are using ("Let \( m \) and \( n \) be integers...")
2. Represent the two odd numbers using those variables ("They can be written as \( 2m+1 \) and \( 2n+1 \)")
3. Perform the calculation ("\( (2m+1) + (2n+1) = 2m + 2n + 2 \)")
4. Transform it into the target form (as \( 2 \times \) integer) ("\( 2(m + n + 1) \)")
5. State the conclusion ("Therefore, the sum is an even number")
◎ Summary of this section:
・Explanation problems follow a set "template"!
・Always transform your final answer into a form that clearly shows its characteristics, like "\( 2 \times (\dots) \), so it's an even number!"
5. Rearranging Formulas
Lastly, we will practice rewriting formulas into the form of "specific variable = ...".
This follows the exact same rules as solving an equation.
Step-by-Step Method
Example: Solve \( 3x + y = 10 \) for \( y \).
(The goal is to get it into the form \( y = \dots \))
- Move the term that doesn't contain \( y \) (\( 3x \)) to the other side (transpose).
\( y = 10 - 3x \)
And that's it!
Example: Solve \( L = 2\pi r \) for \( r \).
1. It's easier to see if you swap the left and right sides.
\( 2\pi r = L \)
2. Divide both sides by the thing bothering \( r \), which is \( 2\pi \).
\( r = \frac{L}{2\pi} \)
【Advice】
When asked to "solve for a variable," your goal is to isolate that variable. Just move everything else to the other side!
◎ Summary of this section:
・Change the sign when you move a term to the other side (transposition).
・If it's connected by multiplication, pull it away using division!
Great work! You've mastered the basics of "Algebraic Calculations."
You might make calculation mistakes at first, but that's a path everyone walks. If you make it a habit to check, "Did I miss any signs?" or "Did I forget to write down a variable?", you will definitely become an expert.
Next, we'll use these expressions to learn a more interesting tool called "Systems of Equations." Are you ready? Let's do this!