Unit 4: Extending the Number System Concept Sheet
Standard: MGSE9-12.N.RN.2 Rewriting expressions involving radicals.
Lesson 01: Simplifying Square Roots
Definition of a Square Root:
Parts of a Radical
In math talk: For any numbers a and b, If
a2=b, then a is a square root of b.
In other words, square root just means 'what
number times itself'.
If 132=169, then 13 is the square root of 169.
Simplifying Square Roots
When simplifying square roots remember to
always look for a perfect square factor.
Use “factoring trees” to simplify radicals
that don’t contain perfect squares.
Lesson 02: Combining Square Roots
Adding and Subtracting Radicals
Steps: 1. Simplify each term.
2. Combine the coefficients of like radicals.
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Multiplying Radicals
Steps: 1. Put all lead coefficients together in front of a radical
(with multiplication).
2. Put all the radicands under one radical
(with multiplication).
Standard: MGSE9-12.N.RN.3 Rational vs. Irrational Properties
Lessons 03-04: Rational and Irrational Numbers
A RATIONAL number is a number that can be expressed as a fraction (or a ratio) of two
integers.
Examples:
An IRRATIONAL number CANNOT be expressed as a fraction. Irrational numbers can
include non-repeating, non-terminating (never stops) decimals; square roots of non-perfect
squares, etc........ the most common 'look' of an irrational number is a number with a radical
in it!
Examples:
KEY STATEMENTS TO ALWAYS REMEMBER! Be able to explain each!
1.
THE SUM OR PRODUCT OF RATIONAL NUMBERS IS RATIONAL.
2.
THE SUM OF A RATIONAL NUMBER AND AN IRRATIONAL NUMBER IS
IRRATIONAL.
3.
THE PRODUCT OF A NONZERO RATIONAL NUMBER AND AN IRRATIONAL
NUMBER IS IRRATIONAL.
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Standard: MGSE9-12.A.APR.1 Arithmetic Operations on polynomials
Lessons 05-06: Polynomial Operations
Definitions

o
A Term is a number, a variable, or the product of a number and variable(s).
For example, the second term in the polynomial 8p + 4r2 +9 is 4r3

o
A Monomial is an expression with a single term.
Examples: x, 6, 5xy, -2xy3

A Polynomial is a monomial, or a sum of monomials. Each monomial or "term" in the
polynomial is different.

o
A Binomial is an expression with 2 terms that are added or subtracted.
Examples: x+3, 2x2 + 5

o
A Trinomial is an expression with 3 terms.
Examples: x2 -2x+3, 3x2 +2xy-4y2

Like Terms are terms that contain the same variables raised to the same power.
Numbers are like terms; for example 1 and 3.4 are like terms. The terms 2x 3 and -4x3 are
also like terms.

Constants are numbers with no variables attached to them. For example, in the
polynomial x+2, 2 is the constant term.

Coefficients are numbers attached to the variables. For example, in the term 4x,
the coefficient is the 4. In the term -6x2, the coefficient is -6.
Please Note: A polynomial can have constants, variables, and exponents, but never division
by a variable. (2/x)
A polynomial cannot have negative exponents.
Example of a Polynomial:
5xy2 - 3x + 5y3 - 3
Examples of Non-Polynomials:
3xy-2, 2/(x+2)

Degree of a polynomial is the largest exponent. For example, 4x 3+8x2+1x+5, the
degree=3.

Standard form: When the terms of a polynomial are arranged from the largest
exponent to the smallest exponent in decreasing order. For example, 4x3+8x2+1x+5 is in
standard form.
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Adding and Subtracting Polynomials
To add or subtract a polynomial, we combine LIKE TERMS by combining the coefficients of
the terms. We never change the exponents on the variables!
For example, when we add 2x2 and -5x2, we use the Distributive property to combine only
the 2 and the -5. Essentially, we ask ourselves "how many x 2's do we have?"
so 2x2 + (-5x2) = (2-5)x2 = -3x2
Remember!!! If we are subtracting a polynomial in parentheses, we must distribute the
negative to everything inside the parentheses before we start combing like terms!
Multiplying Polynomials
To multiply polynomials, we multiply each term in the first polynomial by each term in the
second polynomial (use the Distributive property), then combine like terms if there are any.
For example: a(b + c + d)
= a(b) + a(c) + a(d)
(a + b)(c + d)
= a(c + d) + b(c + d)
= ac + ad + bc + bd
(this is sometimes called the FOIL method)
(a + b)(c + d + e)
= a(c + d + e) + b(c + d + e)
= ac + ad + ae + bc + bd+ be
Remember, when you multiply:

signs.
Be very careful when distributing negative numbers! it's easy to forget to change

Since we're multiply each term, we multiply the coefficients, AND we multiply the
variables, so unlike addition, the exponents on variables will change during the multiplication
step.
*Modeling Polynomials Activity type questions will also be included. Carefully Review
your Task Card.
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