Abeka Algebra 2 Test 12 Flashcards

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Questions and Answers

What is the factorization of $x^6 + y^6$?

  • (x+y)(x^2-xy+y^2)(x-y)(x^2+xy+y^2) (correct)
  • x^2 + y^2
  • (x+y)(x^2+y^2)(x-y)
  • (x^2+y^2)(x^2-2xy+y^2)

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 $f(x)=\frac{x-12}{x-2}$?

<p>All real numbers except 2</p> Signup and view all the answers

If y is inversely related to x and y=3 when x=4, what is y when x=1?

<p>12</p> Signup and view all the answers

What is the reduced form of the equations $2x + 3y - 7z = 5$, $x + y - 2z = 11$, $4x - 10x + 15z = 7$ using ERO?

<p>$x + y - 2z = 11; 2x + 3y - 7z = 5; 4x - 10y + 15z = 7$</p> Signup and view all the answers

What is the profit equation represented by 50 baseball cards, 20 football cards, and 55 card slots?

<p>p=10x+8y</p> Signup and view all the answers

What is the resulting dimensions when multiplying a 3x4 matrix by a 4x2 matrix?

<p>3x2</p> Signup and view all the answers

What is the determinant of the matrix $\begin{bmatrix} 1 & 5 \ 3 & -1 \end{bmatrix}$?

<p>(-1)(1) - (5)(3) = -1 - 15 = -16</p> Signup and view all the answers

What is the Cramer's rule formula for x?

<p>x=|x|/|A|</p> Signup and view all the answers

What does the number line |x-4| represent?

<p>|x-4|</p> Signup and view all the answers

Flashcards

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

Complex numbers are represented as a + bi, where a is the real part and b is the imaginary part.

Quadratic Transformation

Substituting u for x^2 in the equation transforms it, simplifying solving processes.

Solutions via Substitution

From (u - 2)(u - 8) = 0, u = x^2, we find x = ±√2 and x = ±2√2.

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Domain of Rational Functions

The function f(x) = (x - 12)/(x - 2) has a domain excluding x = 2.

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Inverse Variation Calculation

Using y = k/x, if y = 3 when x = 4, find y when x = 1, yielding y = 12.

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Elementary Row Operations (ERO)

ERO simplifies systems like 2x + 3y - 7z = 5 and x + y - 2z = 11 to find equivalences.

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Matrix Multiplication Dimensions

Multiplying a 3x4 matrix by a 4x2 matrix results in a 3x2 matrix.

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Calculating Determinants

For matrix with rows [1, 5] and [3, -1], determinant is -16.

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Cramer's Rule Application

Cramer's rule uses determinants to solve equations for variables x and y.

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Interpretation of Absolute Values

|x - 4| measures distance from 4 on the number line.

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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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