Algebra 2: Expressions Quiz

Study Notes

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.

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.

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 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.

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).

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).