A.2ab and *A.2ab - Polynomials

A.2    The student will perform operations on polynomials, including

a)  applying the laws of exponents to perform operations on expressions;

b)  adding, subtracting, multiplying, and dividing polynomials;

A.2  The student will perform operations on polynomials, including

a)  applying the laws of exponents to perform operations on expressions;

b)  adding, subtracting, multiplying, and dividing polynomials; 

Bloom's Level:  Apply

• I can design the track of a roller coaster, determine the sloped curve of an exit ramp, and understand the mechanics of controllers like autopilot, cruise control, and a living room thermostat.
  

• I will apply mathematical processes for basic operations to algebraic expressions and divide algebraic terms out of algebraic expressions.
  

• The laws of exponents can be investigated using inductive reasoning. 
  
• A relationship exists between the laws of exponents and scientific notation.
  
• Operations with polynomials can be represented concretely, pictorially, and symbolically.
  
• Polynomial expressions can be used to model real-world situations.
  
• The distributive property is the unifying concept for polynomial operations.
  
• Factoring reverses polynomial multiplication.
  
• Some polynomials are prime polynomials and cannot be factored over the set of real numbers.
  
• Polynomial expressions can be used to define functions and these functions can be represented graphically.
  
• There is a relationship between the factors of any polynomial and the x-intercepts of the graph of its related function.
  

·  Operations with polynomials can be represented concretely, pictorially, and symbolically.

·  Polynomial expressions can be used to model practical situations.

·  Factoring reverses polynomial multiplication.

·  Trinomials may be factored by various methods including factoring by grouping.

­  Example of factoring by grouping
2x² + 5x – 3
2x² + 6x – x – 3
2x(x + 3) – (x + 3)
(x + 3)(2x – 1)

·  Prime polynomials cannot be factored over the set of integers into two or more factors, each of lesser degree than the original polynomial.

·  Polynomial expressions can be used to define functions and these functions can be represented graphically.

·  The laws of exponents can be applied to perform operations involving numbers written in scientific notation.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

A.2a₁  Simplify monomial expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

·  A.2a₁  Simplify monomial expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents.

A.2b₁  Model sum and difference of polynomials w/concrete objects and their related pictorial representations.

A.2b₁  Model products of polynomials w/ concrete objects  and their related pictorial representations.

A.2b₁  Model quotients of polynomials w/ concrete objects and related pictorial representations.

·  A.2b₁  Model sums, differences, products, and quotients of polynomials with concrete objects and their related pictorial and symbolic representations.

A.2b₂  Relate concrete and pictorial manipulations that model polynomial operations to their corresponding symbolic representations.

A.2b₃  Find sums and differences of polynomials.

·  A.2b₂  Determine sums and differences of polynomials.

A.2b₄  Find the products of polynomials. The factors will have no more than five total terms.

·  A.2b₃  Determine products of polynomials.  The factors should be limited to five or fewer terms (i.e.,  (4x + 2)(3x + 5) represents four terms and (x + 1)(2x²  + x + 3) represents five terms).

A.2b₅  Find the quotient of polynomials, using a monomial or binomial divisor, or a complete factored divisor.

·  A.2b₄  Determine the quotient of polynomials, using a monomial or binomial divisor, or a completely factored divisor. 

monomial expression, ratio, exponents, integers, sums, difference, product, quotient, polynomial, operations, concrete, pictorial, factors, factor, binomial, divisor, degree, integral coefficients, prime, x-intercepts, graphical representation