Introduction to Algebra
Algebra is the foundation of higher mathematics and an essential skill for every high school student. It provides the tools to describe relationships between quantities, solve problems systematically, and develop logical thinking skills that extend far beyond the mathematics classroom.
At its core, algebra uses letters and symbols to represent numbers in equations and expressions. This abstraction allows us to write general rules that apply to infinitely many specific cases. For example, instead of saying "if I add 5 to some number and get 12, the number is 7," we can write the equation x + 5 = 12 and solve for x.
The Language of Algebra
In algebra, we frequently encounter expressions like:
where x represents an unknown value we need to find.
Linear Equations
A linear equation is an equation where the highest power of the variable is 1. The general form is ax + b = c, where a, b, and c are constants, and a ≠ 0.
Solving Linear Equations
To solve a linear equation, we use the principle that whatever operation we perform on one side of the equation, we must also perform on the other side to maintain equality.
Example 1: Simple Linear Equation
Solve for x: 4x - 3 = 13
Step 1: Add 3 to both sides
4x - 3 + 3 = 13 + 3
4x = 16
Step 2: Divide both sides by 4
4x ÷ 4 = 16 ÷ 4
x = 4
Verification: 4(4) - 3 = 16 - 3 = 13 ✓
Systems of Linear Equations
When we have two or more linear equations with the same variables, we have a system of equations. There are three main methods to solve these:
- Graphing method: Plot both equations and find where they intersect
- Substitution method: Solve one equation for a variable and substitute into the other
- Elimination method: Add or subtract equations to eliminate a variable
Example 2: Solving by Substitution
Solve the system:
y = 2x + 1 ... (1)
3x + y = 11 ... (2)
Step 1: Substitute (1) into (2):
3x + (2x + 1) = 11
Step 2: Solve:
5x + 1 = 11
5x = 10
x = 2
Step 3: Find y from (1):
y = 2(2) + 1 = 5
Solution: x = 2, y = 5
Polynomials
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The degree of a polynomial is the highest power of the variable.
Polynomial Types by Degree
- Degree 0: Constant (e.g., 7)
- Degree 1: Linear (e.g., 3x + 2)
- Degree 2: Quadratic (e.g., x² - 4x + 3)
- Degree 3: Cubic (e.g., 2x³ + x² - 5x + 1)
Operations with Polynomials
Addition and Subtraction: Combine like terms (terms with the same variable raised to the same power).
Example 3: Adding Polynomials
Add: (3x² + 2x - 5) + (x² - 3x + 4)
= 3x² + x² + 2x - 3x - 5 + 4
= 4x² - x - 1
Multiplication: Use the distributive property (FOIL for binomials).
Example 4: Multiplying Binomials (FOIL)
Multiply: (x + 3)(x - 2)
First: x × x = x²
Outer: x × (-2) = -2x
Inner: 3 × x = 3x
Last: 3 × (-2) = -6
= x² - 2x + 3x - 6 = x² + x - 6
Factoring
Factoring is the process of breaking down a polynomial into products of simpler polynomials. It's essential for solving quadratic equations and simplifying expressions.
Common Factoring Techniques
1. Factoring out the GCF (Greatest Common Factor):
Factor: 6x³ + 9x² - 3x
GCF = 3x
= 3x(2x² + 3x - 1)
2. Difference of Squares:
Factor: x² - 16
= x² - 4² = (x + 4)(x - 4)
3. Trinomial Factoring:
Factor: x² + 5x + 6
Find two numbers that multiply to 6 and add to 5: 2 and 3
= (x + 2)(x + 3)
Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. There are several methods to solve quadratic equations:
Method 1: Factoring
Solve: x² - 5x + 6 = 0
Factor: (x - 2)(x - 3) = 0
x - 2 = 0 or x - 3 = 0
x = 2 or x = 3
Method 2: Quadratic Formula
This formula works for ALL quadratic equations.
Example 5: Using the Quadratic Formula
Solve: 2x² + 5x - 3 = 0
a = 2, b = 5, c = -3
x = (-5 ± √(25 - 4(2)(-3))) / 2(2)
x = (-5 ± √(25 + 24)) / 4
x = (-5 ± √49) / 4
x = (-5 ± 7) / 4
x = 1/2 or x = -3
The Discriminant
The expression b² - 4ac under the square root is called the discriminant. It tells us about the nature of the solutions:
- Positive: Two distinct real solutions
- Zero: One repeated real solution
- Negative: Two complex solutions
Functions in Algebra
A function is a relation where each input (x-value) produces exactly one output (y-value). We often write functions as f(x), which means "f of x."
Function Notation
If f(x) = 2x² - 3x + 1, then:
- f(2) = 2(4) - 3(2) + 1 = 8 - 6 + 1 = 3
- f(-1) = 2(1) - 3(-1) + 1 = 2 + 3 + 1 = 6
Function Operations
Composition of Functions: If f and g are functions, then (f ∘ g)(x) = f(g(x))
Example 6: Function Composition
If f(x) = 2x + 3 and g(x) = x - 1, find f(g(x)).
f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x - 2 + 3 = 2x + 1
Key Takeaways
- Linear equations can be solved by isolating the variable using inverse operations
- Systems of equations can be solved by graphing, substitution, or elimination
- Polynomials can be added, subtracted, and multiplied using distributive properties
- Factoring breaks down polynomials into simpler factors
- The quadratic formula solves any quadratic equation
- Functions map inputs to outputs and can be composed, added, and multiplied
Practice What You've Learned
Test your algebra skills with our interactive practice tests featuring randomized questions and detailed explanations.
Take Algebra Test →