Square Roots (√)
A square root is a number that, when multiplied by itself, results in the given value.
Example:
√25 = 5 because 5 × 5 = 25
Perfect Squares
Numbers that have whole-number square roots.
Common Perfect Squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Example:
√64 = 8 because 8 × 8 = 64
Irrational Numbers
A number that cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating.
Examples:
- √2 = 1.414213… (never-ending, non-repeating)
- π = 3.1415926535…
- √7, √5, and e
Estimating Square Roots
If a number is not a perfect square, estimate its square root between two known perfect squares.
Example:
√50
Since 49 < 50 < 64,
√50 ≈ 7.1
(because √49 = 7 and √64 = 8)
Higher-Order Roots (Cube Roots, Fourth Roots, etc.)
Cube Roots (∛)
A cube root is a number that, when multiplied by itself three times, results in the given value.
Example:
∛27 = 3 because 3 × 3 × 3 = 27
Finding the fourth, fifth, or higher-order roots follows the same pattern as squares and cubes.
Example:
⁴√81 = 3 because 3 × 3 × 3 × 3 = 81
Polynomials & Algebraic Expressions
Polynomial Basics
A polynomial is an algebraic expression made up of terms with variables, coefficients, and exponents.
Example of a Polynomial:
5x³ – 2x² + 4x – 7
Polynomial Terms:
- Coefficient: The number in front of a variable (e.g., 5 in 5x³)
- Variable: The letter representing an unknown number (e.g., x)
- Exponent: The power to which the variable is raised (e.g., ³ in 5x³)
- Constant: A number with no variable (e.g., -7)
Combining Like Terms
Only terms with the same variables and exponents can be combined.
Example:
3x² + 5x – 7 + 2x² – 3x + 4
Combine like terms:
(3x² + 2x²) + (5x – 3x) + (-7 + 4)
Final result: 5x² + 2x – 3
Multiplying Polynomials
Using the Distributive Property (Handshaking Method)
Multiply each term in the first expression by each term in the second expression.
Example:
3x(2x + 4)
Distribute 3x:
(3x × 2x) + (3x × 4) = 6x² + 12x
Multiplying Binomials (FOIL Method)
Use the FOIL method (First, Outer, Inner, Last) for multiplying two binomials.
Example:
(5x + 3)(2x – 4)
Multiply:
- First: 5x × 2x = 10x²
- Outer: 5x × -4 = -20x
- Inner: 3 × 2x = 6x
- Last: 3 × -4 = -12
Combine like terms:
10x² – 20x + 6x – 12 = 10x² – 14x – 12
Using a Grid for Polynomial Multiplication
A multiplication grid helps organize terms and ensure correct calculations.
Example: Multiply (x + 2)(x + 3) using a grid:
| x | 3 | |
| x | x² | 3x |
| 2 | 2x | 6 |
Now, add the terms:
x² + 3x + 2x + 6 = x² + 5x + 6
Final Review
- Square Roots: A number that, when squared, gives the original value.
- Perfect Squares: Whole numbers that have whole-number square roots.
- Irrational Numbers: Square roots of non-perfect squares are non-repeating decimals.
- Estimating Square Roots: Find the closest perfect squares to approximate values.
- Cube & Higher-Order Roots: Numbers that multiply themselves multiple times to form the original value.
- Polynomial Basics: Algebraic expressions with variables, coefficients, and exponents.
- Combining Like Terms: Merge terms with identical variables and exponents.
- Multiplying Polynomials: Use distributive property, FOIL, or a grid for accuracy.