Operations with Monomials and Polynomials: Multiplying, and Dividing.

In the previous lesson you had the opportunity to practice arithmetic operations with polynomials up to multiplying them. Now in this lesson the opportunity will be in diving them, including an introduction to synthetic division that will be further develped when working to find solutions in polynomial equations later in this site.

The lesson finilizes with expanding and factoring polynomials. These are mainly difference of squares, perfect squares, and difference of cubes. There is an extensive use of algebra tiles to model the operations, and for most of the problems, you will have the option to solve them right there in the screen using the MARKER TOOLS MENU and the provided pen and highlighter. For the ones that may be done by modeling aggregation of rectangular areas to represent the individual quadratic, lineal and independent terms; you are provided with an unobstructive grey out grid you may use to guide your drawing.

Lesson's Content

Lesson In PDF Format (no animations)

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Lesson's Glossary

Algebraic expression: A given set of letters called variables, and real numbers called constants that are combined using addition, subtraction, multiplication, division and/or exponentiation.

Associative Property: Grouping addends in a sum or factors in a product in different order does not affect the answer.

Binomial: Expression that has two (bi) terms.

Commutative Property: Modifying the order of addends or factors in a expression does not affect the sum or product.
Order is not important when adding or multiplying.

Constant: It is a number or value that remains always the same. Never changes.

Difference of two squares: a2 - b2 = (a +b)(a - b)

Exponent: It is a raised number representing the repeated multiplication of a given factor.

Perfect square trinomial: A trinomial generated by the product of two equal binomials.

Expression: Any combination of numbers and operations without the = sign.

Factoring: The process to brake a polynomial down into the product of several factors.

Factors: All whole numbers that are multiplied together to yield another number.

Factored Form: Any polynomial that is written as the product of polynomials of lower degree that may be obtained from the original polynomial.

Like terms: Terms that have the same combination of variables to the same power as factors.

Operation: Any action we perform on one or two numbers to produce a new number. Most common ones are addition, subtraction, multiplication, division, square roots, powers, and so on.

Polynomial: A algebraic statement with one or more terms. Word comes from “poly” which means many.

Polynomial (third degree): Geometric representation. A third degree polynomial may be represented as the volume of a rectangular prism for which length, width and depth are the linear factors of the polynomial.

Power: Exponent of a number or variable.

Subtraction: Adding the opposite.

Synthetic division: A process of dividing a polynomial by a linear factor, using the coefficients and ignoring the variable and the exponents (they are relevant just as the position to place the coefficients). When dividing the sign of the divisor is reversed to avoid subtracting and to allow just to add.

Term: A form of grouping one or more numerical and/or variable factors by means of multiplication and division. Addition and subtraction symbols separate terms.

Variable: A letter used to represent a number. When the variable is part of an equation, it is possible to find the value for which it stands for by solving the equation. This is the solution (s) of the equation.

Variable expression: Mathematical phrase with at least one variable in it.

 

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