Questions and Answers
What is the factorization of $x^6 + y^6$?
What is the standard form of complex numbers represented by 8 - 4i?
8 - 4i
What is the result of applying U substitution to $x^2 - 10x^2 + 16 = 0$?
(u-2)(u-8)=0; x=±√2 and x=±2√2
What is the domain of the function $f(x)=\frac{x-12}{x-2}$?
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If y is inversely related to x and y=3 when x=4, what is y when x=1?
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What is the reduced form of the equations $2x + 3y - 7z = 5$, $x + y - 2z = 11$, $4x - 10x + 15z = 7$ using ERO?
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What is the profit equation represented by 50 baseball cards, 20 football cards, and 55 card slots?
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What is the resulting dimensions when multiplying a 3x4 matrix by a 4x2 matrix?
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What is the determinant of the matrix $\begin{bmatrix} 1 & 5 \ 3 & -1 \end{bmatrix}$?
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What is the Cramer's rule formula for x?
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What does the number line |x-4| represent?
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Study Notes
Algebra Concepts and Definitions
- Factoring Sum of Cubes: The expression ( x^6 + y^6 ) factors into ( (x+y)(x^2 - xy + y^2)(x-y)(x^2 + xy + y^2) ).
- Standard Form of Complex Numbers: A complex number can be represented as ( 8 - 4i ), where 8 is the real part and -4 is the imaginary part.
Quadratic Equations and Substitution
- Quadratic Transformation: For the equation ( x^2 - 10x^2 + 16 = 0 ), using ( u ) substitution transforms it to ( u^2 - 10u + 16 = 0 ).
- Solutions via Substitution: From ( (u - 2)(u - 8) = 0 ), ( u = x^2 ) leads to solutions ( x^2 = 2 ) (giving ( x = \pm \sqrt{2} )) and ( x^2 = 8 ) (giving ( x = \pm 2\sqrt{2} )).
Functions and Their Domains
- Domain of Rational Functions: The domain of the function ( f(x) = \frac{x - 12}{x - 2} ) includes all real numbers except ( x = 2 ) due to division by zero.
Inverse Relationships
- Inverse Variation Calculation: If ( y = 3 ) when ( x = 4 ), finding ( y ) when ( x = 1 ) yields ( y = 12 ) through inverse relationship calculations.
Systems of Equations
- Elementary Row Operations (ERO): System of equations represented as ( 2x + 3y - 7z = 5 ) and ( x + y - 2z = 11 ) is simplified to identify equivalent equations.
- Matrix Consistency: The equivalent forms of equations highlighted include ( x + y - 2z = 11 ), ( 2x + 3y - 7z = 5 ), and ( 4x - 10y + 15z = 7 ).
Combinatorics and Profit Function
- Card Counting Problem: With 50 baseball cards and 20 football cards, the profit function can be expressed as ( p = 10x + 8y ), where ( x ) and ( y ) represent the quantities of each card type.
Matrix Operations
- Matrix Multiplication Dimensions: The product of a ( 3 \times 4 ) matrix and a ( 4 \times 2 ) matrix yields a resulting matrix of size ( 3 \times 2 ).
Determinants and Cramer's Rule
- Calculating Determinants: The determinant of the matrix formed by rows ( [1, 5] ) and ( [3, -1] ) results in ( (1 \times -1) - (3 \times 5) = -1 - 15 = -16 ).
- Cramer's Rule Application: Using Cramer’s rule for calculating variables ( x ) and ( y ), ( x ) is determined by the ratio of the absolute value of determinants ( |x| ) to the determinant of matrix ( |A| ).
Absolute Value and Inequalities
- Interpretation of Absolute Values: The expression ( |x - 4| ) represents the distance from the number 4 on a number line.
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Prepare for the Abeka Algebra 2 Test 12 with these flashcards. Each card covers key concepts such as complex numbers, function domains, and polynomial expressions crucial for mastering algebra. Test your skills and understanding as you review vital algebraic principles.
