# 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

### BIG IDEAS

• 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.

### UNDERSTANDING THE STANDARD

• 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 + 6xx – 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.

·  For division of polynomials in this standard, instruction on the use of long or synthetic division is not required, but students may benefit from experiences with these methods, which become more useful and prevalent in the study of advanced levels of algebra.

### ESSENTIALS

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.2aSimplify 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.2bModel 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.2bDetermine sums and differences of polynomials.

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

·  A.2bDetermine 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)(2x+ x + 3) represents five terms).

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

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

### KEY VOCABULARY

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

Updated: Oct 27, 2017