Abeka Algebra 2 Test 12 Flashcards

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.