PolynomialsPolynomialsPolynomialsPolynomialsToday’s Objectives:

Identify, Add & Subtract Polynomials

Today’s Objectives:

Identify, Add & Subtract Polynomials

What you should alreadyalready know

What you should alreadyalready know

Monomials Constants, Coefficients, & Degrees

Properties of Powers Writing Differences as Sums The Distributive Property

Monomials Constants, Coefficients, & Degrees

Properties of Powers Writing Differences as Sums The Distributive Property

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z

€

w

€

−3t 4

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3x

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x+ y

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2wc3 + x+ y2

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xy −12

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x12

Monomial? # of terms? Polynomial? Binomial? Trinomial?

PolynomialsPolynomials A polynomial is a monomial or sum of

monomials The monomials in a polynomial are

called the terms of the polynomial. x2+2x+1 has 3 terms: x2,2x, & 1

Binomials: Have 2 unlike terms Trinomials: Have 3 unlike terms

A polynomial is a monomial or sum of monomials

The monomials in a polynomial are called the terms of the polynomial. x2+2x+1 has 3 terms: x2,2x, & 1

Binomials: Have 2 unlike terms Trinomials: Have 3 unlike terms

Determine whether each expression is a polynomial. If it is a polynomial, how many terms are there and state the degree of the polynomial. If it is not state

why.

Determine whether each expression is a polynomial. If it is a polynomial, how many terms are there and state the degree of the polynomial. If it is not state

why.Polynomial? How many Terms?Degrees?

Polynomial? How many Terms?Degrees?

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1

6x 3y 5 − 9x 4

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x + x + 5Polynomial? How many Terms?Degrees?

Polynomial? How many Terms?Degrees?

Using Algebra TilesUsing Algebra Tiles

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2x 2 + 4x + 2

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x 2 − 2x + 3

Using Algebra Tiles:Simplify:

Using Algebra Tiles:Simplify:

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(3x 2 + 5x +1) + (4x 2 − 6x + 7)

Using Algebra Tiles:Simplify:

Using Algebra Tiles:Simplify:

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(−a2 + 3a+ 6) + (5a2 + 4a− 2)

Example: Simplify: Example: Simplify:

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(y 3 + 6y 2 + y −1) + (−2y 3 + y 2 − 6y +10)

Example: Simplify: Example: Simplify:

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(x 2 + 5x + 7) + (x 2 − 6x − 2) + (x 2 + 2x + 3)

Using Algebra Tiles:Simplify:

Using Algebra Tiles:Simplify:

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(5x 2 + 2x + 8) − (3x 2 + x + 4)

Example:Simplify

Example:Simplify

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(5x 3 + 2x 2 − 4x) − (3x 3 − 4x 2 + 8)

YOUR TURN!YOUR TURN!

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(−4m2 − 6m) − (6m + 4m2)

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17 j 2 −12k 2 + 3 j 2 −15 j 2 +14k 2

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18p2 +11pq− 6q2 − (15p2 − 3pq+ 4q2)

A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool.

A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool.

MonomialsMonomials A monomial is an expression that is a

number, a variable, or the product of a number and one or more variables.

A monomial cannot contain variables in denominators, variables with exponents that are negative, or variables under radicals.

Constants Coefficients Degrees

A monomial is an expression that is a number, a variable, or the product of a number and one or more variables.

A monomial cannot contain variables in denominators, variables with exponents that are negative, or variables under radicals.

Constants Coefficients Degrees

Essential Characteristics Nonessential Characteristics

Example Non-Example

Monomials

ConstantsConstants

Monomials that contain no variables,

Example: 23 or -1

Monomials that contain no variables,

Example: 23 or -1

CoefficientCoefficient

The numerical factor of a monomial

Example: -6m-6 is the coefficient

The numerical factor of a monomial

Example: -6m-6 is the coefficient

DegreeDegree Sum of the variables’ exponents. Where there

are multiple terms; the highest sum of the exponents among each term.

Examples: 12g7h4

7+4=11

3ax6+12y7z2

6+1=7, 7+2=9the degree is 9

Sum of the variables’ exponents. Where there are multiple terms; the highest sum of the exponents among each term.

Examples: 12g7h4

7+4=11

3ax6+12y7z2

6+1=7, 7+2=9the degree is 9

Properties of PowersProperties of Powers

Power of a Power: (am)n=amn

Power of a Product: (ab)m=ambm

Power of a Quotient: (a/b)n = (an/bn); b 0 and (a/b)-n = (b/a)n = (bn/an); a 0, b 0

Power of a Power: (am)n=amn

Power of a Product: (ab)m=ambm

Power of a Quotient: (a/b)n = (an/bn); b 0 and (a/b)-n = (b/a)n = (bn/an); a 0, b 0

Writing Differences as SumsWriting Differences as Sums

2-7 = 2+(-7)

-6-11 = -6+(-11)

x-y = x+(-y)

2-7 = 2+(-7)

-6-11 = -6+(-11)

x-y = x+(-y)

8-2x = 8+(-2x)

2xy-6yz = 2xy+(-

6yz) 6a2b-12b2c =

6a2b+(-12b2c)

8-2x = 8+(-2x)

2xy-6yz = 2xy+(-

6yz) 6a2b-12b2c =

6a2b+(-12b2c)

The Distributive PropertyThe Distributive Property

-2(4x3+x-30)=-8x3-

2x+6 -1(x+2)=

-x-2 (-1/2)(3a+2)=

(-3/2)a-1

-2(4x3+x-30)=-8x3-

2x+6 -1(x+2)=

-x-2 (-1/2)(3a+2)=

(-3/2)a-1

Algebra TilesAlgebra Tiles

x2

x

-x

1

Monomial? # of terms Polynomial?Binomial? Trinomial?

x 1 x

1

x 1 x

x 1 x

1

2 x binomial

2 binomial

3 x trinomial

€

z

€

w

€

−3 t 4

€

3x

€

x + y

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2wc3 + x + y 2

€

xy −12

€

x12