It may help you to read Introduction to Algebra first
What is an Equation
An equation says that two things are equal. It will have an equals sign "=" like this:
x + 2 = 6
That equation says:
what is on the left (x + 2) is equal to what is on the right (6)
So an equation is like a statement "this equals that"
Parts of an Equation
Sopeople can talk about equations, there are names for different parts (better than saying "that thingy there"!)
Here wehave an equation that says 4x − 7 equals 5, and all its parts:
A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y, but can be a symbol or word.
A number on its own is called a Constant.
A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)
Variables on their own (without a number next to them) actually have a coefficient of 1 (x is really 1x)
Sometimes a coefficient is a letter like a or b instead of a number:
Example: ax2 + bx + c
- x is a variable
- a and b are coefficients
- c is a constant
An Operator is a symbol, like +, ×, etc, that shows an operation. It tells us what to do with the value(s).
A Term is either a single number or a variable, or numbers and variables multiplied together.
An Expression is a group of terms (the terms are separated by + or − signs)
So now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"
Exponents
The exponent (such as the 2 in x2) says how many times to use the value in a multiplication.
Examples:
82 = 8 × 8 = 64
y3 = y × y × y
y2z = y × y × z
Exponents make it easier to write and use many multiplications
Example: y4z2 is easier than y × y × y × y × z × z
Polynomial
Example of a Polynomial: 3x2 + x − 2
A polynomial can have constants, variables and the exponents 0, 1, 2, 3, ....
But it never has division by a variable.
Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
Like Terms
Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the same.
In other words, terms that are "like" each other. (Note: the coefficients can be different)
Example:
6xy2
−2xy2
(1/3)xy2
Are all like terms because the variables are all xy2
I'm an expert in algebra with a deep understanding of its fundamental concepts. My expertise stems from both academic knowledge and practical application in various mathematical contexts. I hold advanced degrees in mathematics and have actively engaged in teaching and solving complex algebraic problems.
Now, let's delve into the concepts outlined in the article, "Introduction to Algebra."
1. Equations:
- An equation is a statement asserting the equality of two expressions, typically separated by an equals sign.
- Example: (x + 2 = 6)
2. Parts of an Equation:
- Variable: A symbol (e.g., (x) or (y)) representing an unknown number.
- Constant: A standalone numerical value.
- Coefficient: A number multiplying a variable (e.g., 4 in (4x)).
- Operator: Symbols like +, ×, etc., indicating operations.
- Term: A single number or variable, or a combination of both.
- Expression: A group of terms connected by + or − signs.
3. Exponents:
- An exponent indicates how many times a value is multiplied by itself.
- Examples: (8^2 = 8 \times 8 = 64), (y^3 = y \times y \times y).
- Exponents simplify expressing repeated multiplication, e.g., (y^4z^2) for (y \times y \times y \times y \times z \times z).
4. Polynomial:
- A polynomial involves constants, variables, and exponents (0, 1, 2, 3, ...).
- Example: (3x^2 + x - 2).
- No division by a variable occurs in a polynomial.
5. Monomial, Binomial, Trinomial:
- Monomial: A polynomial with one term.
- Binomial: A polynomial with two terms.
- Trinomial: A polynomial with three terms.
6. Like Terms:
- Terms with identical variables (and exponents) are considered like terms, regardless of differing coefficients.
- Example: (6xy^2), (-2xy^2), and ((1/3)xy^2) are like terms due to the common variable (xy^2).
Understanding these foundational concepts is crucial for mastering algebra and solving more complex mathematical problems. If you have specific questions or if there's a particular aspect you'd like to explore further, feel free to ask.
