Questions and Answers
Which of the following points lies in the second quadrant?
What are the coordinates of the origin in the Cartesian plane?
Which point lies on the y-axis?
Identify the point that has a negative abscissa and positive ordinate.
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Which of the following points lies on the x-axis?
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What is the factorization of $x^3 + 125$?
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What is the correct factorization of $a^3 + 27b^3$?
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Which identity correctly represents the factorization of $x^3 + 216$?
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What is the factorization for $1 - 8x^3$?
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What factorization is derived from $x^3 + y^3 + 2x^2y$?
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What is the factorization of $a^3 + 276^3$?
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Which expression correctly represents the factorization of $8a^3 + 276^3$?
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What is the factorization of $ab^7 + ba^7$?
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How would you factor the expression $x^3y^3 + 216$?
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What is the correct factorization method for the expression $x^3 + y^3$?
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Study Notes
Algebraic Identities
Factorization
- a³ + 27b³: Can be factored using the sum of cubes identity as (a + 3b)(a² - 3ab + 9b²).
- x³ + 125: Factored as (x + 5)(x² - 5x + 25) based on the sum of cubes identity.
- 8a²b³ + 27b³: Factored as (2b³)(4a² + 27) using a common factor.
- x⁵ + 27x²: Factored using common terms, resulting in x²(x³ + 27) = x²(x + 3)(x² - 3x + 9).
- ab³ + ba²: Factored as ab(a + b) by taking out the common factor ab.
- 8(a + b)³ + 27: Can be expressed as 2(a + b)³ + 3³, leveraging sum of cubes to yield (a + b + 3)(a + b)² - 3(a + b)(3).
- x³ + 216: Factorizes to (x + 6)(x² - 6x + 36) using the sum of cubes.
- 125x² + 1/8: Expressed as (5x + 1/2)(25x² - 5x + 1/4) based on sum of cubes.
- a³ + b³ + a + b: This can be factored using grouping, yielding (a + b)(a² - ab + b² + 1).
- x³ + y³ + 2x²y: Results in (x + y)(x² + y² - xy) upon factoring.
- 1 - 8x³: Factorized as (1 - 2x)(1 + 2x + 4x²) using the difference of cubes.
- 64x³ - 27y³: Factored as (4x - 3y)(16x² + 12xy + 9y²) employing the difference of cubes.
- (a + b)³ - (8)³: Can be factored to (a + b - 2)(a² + b² + 4ab + 2(a + b)).
- x³y³ - 512: Expressed as (xy - 8)(x²y² + 8xy + 64) using the difference of cubes.
- a² - 27/a: Can be rewritten as (a - 3√3)(a + 3√3) considering the difference of squares.
- 8a³/27 - b³/8: Factorized as (2a/3 - b/2)(4a²/9 + ab/3 + b²/4).
Special Factorization Identity
- Identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc).
- Condition: If x + y + z = 0, then factor x³ + y³ + z³ is expressed as 3xyz.
Cartesian System
- Point Determination: The horizontal and vertical lines are called x-axis and y-axis, respectively.
- Plane Sections: The two axes divide the plane into four quadrants.
- Point of Intersection: The origin (0,0) is where the x-axis and y-axis intersect.
Quadrants and Point Locations
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Quadrant Assignments:
- Point A (2, 1): Quadrant I
- Point B (-2, 2): Quadrant II
- Point C (-5, -3): Quadrant III
- Point D (2, -3): Quadrant IV
- Point E (3, -7): Quadrant IV
- Point F (-4, 2): Quadrant II
- Point G (2, 3): Quadrant I
- Point H (-1, -4): Quadrant III
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Axes Identification:
- Points on x-axis: B (0,3), C (3,0), D (0,-1), E (-1,0), F (0,0), G (-5,0), H (0,-6).
- Points on y-axis: B (0, 3), D (0, -1), F (0, 0), H (0, -6).
Plotting Points
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Coordinates:
- Point A (-3, 5)
- Point B (3, -5)
- Point C (-2, -4)
- Point D (5, -5)
- Point E (6, -7)
- Point F (0, 4)
- Point G (5, 0)
- Point H (-1, -3)
- Graphical Representation: Understanding how to represent these points on the Cartesian plane is crucial for visualizing relationships between values.
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Description
Test your knowledge of algebraic identities with this quiz. It includes a variety of factorization problems as well as questions based on specific cubic identities. Perfect for students looking to reinforce their understanding of algebraic expressions.
