Questions and Answers
What is the expression for the difference of 5 times the cube of x cubed and the quotient of 4 times x and 3?
5x^3 - 4x/3
What is the expression for 5 times the cube of x divided by 4 times x?
5x^3 / 4x
What is the expression for the quotient of the difference of 5 times x cubed and 4 and x?
5x^3 - 4 / x
What is the expression for the cube of the difference of 5 times x and 4?
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What is the expression for the product of x and the sum of 6 and 8 times the square of x?
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What does the factor 4x + 1 represent in the expression for the volume of a cylindrical soap dispenser?
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What does the constant term in the expression 180x/(x+4) + 250 represent?
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How can the expression x^2 + 3x be factored?
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What is a polynomial?
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What is a term in algebra?
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What is a coefficient in algebra?
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Provide an example of a polynomial with its coefficient.
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What are the factors of 14 related to the expression x^2 + 9x + 14?
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What is 4x^2 + 14x + 6 factored?
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Study Notes
Algebraic Expressions and Definitions
- Difference of Expressions: 5x³ - (4x/3) represents the difference between five times the cube of x cubed and the quotient of four times x divided by three.
- Division of Expressions: 5x³ / (4x) describes five times the cube of x divided by four times x.
- Quotient of a Difference: (5x³ - 4) / x indicates the quotient formed by subtracting four from five times x cubed and then dividing by x.
- Cube of a Difference: (5x - 4)³ portrays the cube of the expression resulting from the difference between five times x and four.
- Product of Variables and Expressions: x(6 + 8x²) illustrates multiplying x by the sum of six and eight times the square of x.
Geometry in Algebra
- Volume of a Cylinder: π(x + 1)²(4x + 1) is an expression modeling the volume of a cylindrical soap dispenser, where 4x + 1 acts as the height.
Real-World Application in Pricing
- Cab Driver Earnings: The expression 180x / (x + 4) + 250 calculates the average fare collected by cab drivers, with 250 as the amount collected when no senior citizens use the cab.
Polynomial Basics
- Polynomial Definition: A polynomial refers to an algebraic expression made up of terms combined via addition or subtraction.
- Definition of a Term: A term can either be a single number, a variable, or a product of numbers and variables, with terms being connected through addition or subtraction.
Coefficients and Factoring
- Coefficient Explained: The coefficient is the numerical factor multiplying a variable in a term, for example, in 2x² + x, the coefficient of x² is 2, and the coefficient of x is 1.
- Factoring Quadratic Expressions: To factor a quadratic expression like x² + 9x + 14, identify two numbers that multiply to the constant term (14) and add up to the linear coefficient (9); this factors as (x + 2)(x + 7).
Grouping in Factoring
- Factoring with Non-Unit Leading Coefficient: For expressions such as 4x² + 14x + 6, first factor out the greatest common factor (2), then use grouping and basic quadratic factorization techniques to obtain the factored form 2(x + 3)(2x + 1).
Studying That Suits You
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Description
Test your knowledge on manipulating and interpreting algebraic expressions with this set of flashcards. Each card presents a word problem related to expressions involving cubes and quotients. Perfect for Algebra 2 students looking to reinforce their understanding of key concepts.
