Algebra Polynomial Quiz

Algebra

• Algebra uses symbols to represent mathematical processes. It evolved from earlier rhetorical approaches, where everything was described in everyday language.
• Ancient civilizations like the Babylonians, Egyptians, and Greeks, and later Arabs, used rhetorical algebra.
• Geometric representations were used to justify algebraic relationships and solve equations, called geometric algebra (e.g., by mathematicians like Pythagoras and Euclid).
• Diophantus (3rd century) introduced abbreviations to simplify algebraic language.
• In the 15th and 17th centuries, mathematical terminology became more symbolic, leading to modern algebraic notation.

Problem Solving

• A problem involving sacks of wheat is presented: A portion was sold, another portion consumed, leaving five sacks remaining. The problem asks for the total number of sacks.
• Another problem involves calculating the number of pellets and sticks needed for triangular structures of increasing size.
• A table in the textbook helps analyze the problem, showing relationships between the number of levels, pellets, and sticks.

Algebraic Expressions and Equations

• An example problem involves scores on a 20-question test, scoring 3 points for each correct answer and losing 2 points for each incorrect/blank answer. The problem aims to determine the number of correct and incorrect responses achieving a score of 40.
• The solutions use variables 'x' (correct answers) and the equation 3x - 2(20-x) = 40, to determine the quantities.
• Further examples demonstrate the use of algebraic equations to solve word problems and express relations between variables (e.g., interest earned by a capital in a bank ).
• An example of a problem-solving process for this test score example is shown: variables for correct answers are identified, variables for incorrect answers are articulated (20 - x) and these answers are plugged into an equation (3x - 2(20 - x) = 40).
• Other formulas/expressions are shown, related to calculations of the area of a rhombus and the calculation of an interest earned.

Generalizing Numerical Series (Term General)

• Examples are given to demonstrate how algebraic expressions can generalize numerical series. A sequence is shown with the terms 0, 2, 6, 12, 20, the general term is shown as (n - 1)n / 2.
• This helps to determine any term in the series using a formula.

Expressions and Operations using Unknowns

• Expressions involving natural numbers (their next term, double of next term, square of next term, etc) are explored.
• This introduces concept of expressions with variables representing unknowns.
• Examples show how to translate word problems and descriptions into algebraic expressions to find the value of an unknown variable.
• This includes translating age relationships in a family into expressions, and solving for an unknown age variable.

Monomials

• Monomials comprise a coefficient (known value) multiplied by literals (unknown values). These literals are often represented as letters.
• The degree of a monomial is determined by the number of literal factors.
• Numerical values are calculated for monomials with particular assigned values.
• Similar monomials are identified with identical literal parts.

Polynomials

• Polynomials are sums (or differences) of monomials. Binomials are sums (differences) of two monomials; Trinomials are sums (differences) of three monomials.
• Degree of a polynomial is the highest degree of its monomials.
• Calculations for the numeric value of polynomials are given.
• The process of defining and calculating the opposite (opposite sign) of a polynomial is demonstrated.

Multiplication and Division of Monomials

• The product of two monomials is another monomial with its degree.
• The division of two monomials can be a number, a monomial or a fraction.
• The properties of numerical operations apply equally to algebraic expressions, including multiplication and division.

Sum and Difference of Polynomials

• Polynomials can be added and subtracted by performing operations on appropriate monomials.
• The examples demonstrate using a method where elements with the same variable expression are grouped together.

Product of Polynomials

• The process for calculating multiplication of polynomials is explained.
• The examples show using the distributive property to multiply all elements in one polynomial with all elements in another polynomial.

Factoring

• The process of factoring is presented as extracting a common factor from a sum/difference of terms.
• Examples demonstrate common factoring from algebraic expressions.

Applications

• The use of algebra in problem situations across different contexts (e.g., calculating interest, calculating test results) is illustrated.

Products of Special Cases (Products Notables)

• Products of special cases of polynomials, like the square of a sum or difference of two monomials, are explored for rapid calculation (e.g., (x + 1)², (x - 4)² etc).
• Simplifications of fractions using the concepts of special cases (e.g., factoring expression like x²-4, and expressing sums of squares by the difference of squares).