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Questions and Answers
What is the formula for the sum/differences of squares?
What is the formula for the sum/differences of squares?
- a² + b² = (a + b)(a - b)
- a² + b² = (a - b)(a + b)
- a² - b² = (a - b)(a - b)
- a² - b² = (a + b)(a - b) (correct)
What do you do with the values 'a' and 'b' in the sum/differences of squares formula?
What do you do with the values 'a' and 'b' in the sum/differences of squares formula?
Square root the 'a' and 'b' values.
Which of the following represents the difference of two cubes?
Which of the following represents the difference of two cubes?
- a³ + b³ = (a - b)(a² + ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²) (correct)
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a + b)(a² - b²)
What is the quadratic formula?
What is the quadratic formula?
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What does the discriminant indicate?
What does the discriminant indicate?
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What can you conclude if the discriminant is less than zero?
What can you conclude if the discriminant is less than zero?
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What does 'i' represent in complex numbers?
What does 'i' represent in complex numbers?
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What is the value of i to the power of 4?
What is the value of i to the power of 4?
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Study Notes
Factoring Formulas
-
Sum/Difference of Squares:
- Formula: ( a² - b² = (a + b)(a - b) )
- To use this formula, find the square roots of 'a' and 'b'.
- Example: For ( 9c² - 4d² ), it factors to ( (3c + 2d)(3c - 2d) ).
-
Sum/Difference of Two Cubes:
- Difference of two cubes: ( a³ - b³ = (a - b)(a² + ab + b²) )
- Steps: Factor out any monomial, then apply the cube formula.
- Example: ( 40c³ - 5d³ ) factors to ( 5(2c - d)(4c² + 2cd + d²) ) after breaking down ( 8c³ ) to ( (2c)³ ).
- Sum of two cubes: ( a³ + b³ = (a + b)(a² - ab + b²) ).
Quadratic Formula
- Quadratic Formula:
- Expressed as ( x = \frac{-b ± √(b² - 4ac)}{2a} ).
- Useful for solving quadratic equations by plugging in values for a, b, and c.
Discriminant
- Discriminant:
- Represents the value inside the square root of the quadratic formula.
- Conditions based on the discriminant:
- Greater than zero: Two real solutions.
- Equal to zero: One real solution.
- Less than zero: No real solutions and two imaginary solutions.
Complex Numbers
- Complex Numbers:
- Defined as ( i = √-1 ).
- Power properties:
- ( i² = -1 )
- ( i³ = -i )
- ( i⁴ = 1 )
- ( i⁵ = i )
- Example for simplification: When dividing a number by 4, if there's a remainder, represent it using ( i⁴ ) and ( i ) multiplied by the remainder value.
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