1. f( g(x)) = ____ for g(x) = 2x + 1 and f(x) = 4x , if x = 3

2. (f + g)(x) = ____ for g(x) = 3x2+ 2x and f(x) = 3x + 1

3. (f/g)(x) = ______ for f(x) = 3x2 +3x and g(x) = 3x

4. (f/g)(x) = ______ for f(x) = 3x2 +3x and g(x) = 3x, for x = 3

Warm-Up

Students will be able to simplify like terms when adding, subtracting, multiplying and dividing polynomials and solving equations.Students will be able to

Objective

Monomial – An expression like 5x is called a monomial. A monomial is an integer, a variable, or a product of integers or variables.

Coefficient – The numerical part of a monomial

Like Terms – When monomials are the same or differ only by their coefficients they are called like terms

Exponent – tells how many times a number, called the base, is used as a factor.

Powers – numbers that are expressed using exponents are called powers.

Multiplicative Inverse/Reciprocals – Two numbers whose product is 1. For example, 2 * ½ = 1, and 2/3 * 3/2 = 1.

Additive Inverse – An integer and its opposite are called additive inverses of each other. The sum of an integer and its additive inverse is zero. For example x + (-x) = 0 and 3 + (-3) = 0.

Polynomial – An algebraic expression that contains one or more monomials is called a polynomial; two term polynomials are called bionomical, and three term polynomials are called trinomial.

Vocabulary and background

Rule – Zero pair is formed by pairing one tile with its opposite or by adding additive inverses.

Simplify

(2x2 + 2x – 4) + (x2 + 3x + 6) 1. Distribute the 1

+

2x2 + 2x – 4 + x2 + 3x + 6 2. Group like terms.

+ +2x2+ x2 + 3x + 2x -4 + 6 3. Add or subtract

coefficients

3x2 + 5x+2 4. Answer

Adding Polynomials

x2x2x2

x x -1

-1-1

-1 xxx 1

11

1

1

1

x2x2x2 x x -1

-1-1

-1xxx

1

1

1

1

1

1

Rule – Zero pair is formed by pairing one tile with its opposite or by adding additive inverses.

Simplify

(2x2 + 2x – 4) – (x2 + 3x + 6) 1. Distribute the -1

-

2x2 + 2x – 4 - x2 - 3x - 6 2. Group like terms,.

+ +2x2 - x2 + 2x - 3x -4 - 6 3. Add or subtract

coefficients

x2 + (-x) + (-10) or x2- x – 10 4. Answer

Subtracting Polynomials-x2 x -xx21-1

x2x2

x x -1

-1-1

-1

-x-x-xx2x2

-1

-1-1

-1

-x2

-1

-1

-1

-1-1

-1x x

x2

xxx 1

11

1

1

1

1

Increasing the Challenge

Add.

1. ( 6y – 5r) + (2y + 7r)

2. (6x2 + 15x – 9) + (5 – 8x – 8x2 )

Subtract

1. (4x2 + 7x + 4) – (x2 + 2x + 1)

2. (5x2y2 + 11xy – 9 ) – ( 9x2y2 – 13xy + 6)

Increasing the Challenge

Add.

1. ( 6y – 5r) + (2y + 7r)

6y – 5r + 2y + 7r Distributive Property / Distribute 1 to remove the parentheses

6y + 2y -5r + 7r Group like terms and simplify

8y + 2r

2. (6x2 + 15x – 9) + (5 – 8x – 8x2 )

6x2 + 15x – 9 + 5 – 8x – 8x2 Distributive Property / Distribute 1 to remove the parentheses

6x2 – 8x2 + 15x – 8x– 9 + 5 Group like terms and simplify

– 2x2 – 7x– 4

Subtract

1. (4x2 + 7x + 4) – (x2 + 2x + 1)

4x2 + 7x + 4 – x2 - 2x – 1 Distributive Property / Distribute -1 to remove the parentheses

4x2 – x2 + 7x - 2x + 4 – 1 Group like terms and simplify

3x2 + 5x + 3

2. (5x2y2 + 11xy – 9 ) – ( 9x2y2 – 13xy + 6)5x2y2 + 11xy – 9 – 9x2y2 + 13xy – 6 Distributive Property / Distribute -1 to remove the parentheses

5x2y2 – 9x2y2 + 11xy + 13xy – 9 – 6 Group like terms and simplify

– 4x2y2 + 23xy – 15

Multiplying and Dividing

Polynomials

Powers of MonomialsProduct of Powers – You can multiply powers that have the same base

by adding

their exponents. For any number a and positive integers m and n.

am * an= a m+n

Ex. a5 * a3 = a5-3 = a2

Quotient of Powers – You can divide powers that have the same base by subtracting

their exponents. For any nonzero number a and whole numbers m and n.

Ex. a4/a2 = a4-2 = a2

Negative Exponents – For any number a and any integer n, a-n =

Ex. 5-2 = 1/52 = 1/25

mm n

n

aa

a

1na

1na

Try This

1. Which expression is equivalent to (4x2 + 8x + 4) – (x2 + 2x + 2).

a. 3x2 + 6x + 2

b. 3x2 + 10x + 6

c. 3x2 - 6x + 2

d. 5x2 + 6x + 2

2

2. Which expression is equivalent to .

a. 2x3y6z3

b. 2xy2z2

c. 4y2z2

d. 4yz3

3 4 2

3 2

4

2

x y z

x y z

Try This

1. Which expression is equivalent to (4x2 + 8x + 4) – (x2 + 2x + 2).

a. 3x2 + 6x + 2

b. 3x2 + 10x + 6

c. 3x2 - 6x + 2

d. 5x2 + 6x + 2

2

2. Which expression is equivalent to .

a. 2x3y6z3

b. 2xy2z2

c. 4y4z2

d. 4yz3

Explanation: (16/4)( x6-6)(y8-4)(z4-2) = 4y4z2

Quotient of Powers

3 4 2

3 2

4

2

x y z

x y z

4x2 + 8x + 4 – x2 - 2x – 24x2 – x2 + 8x - 2x + 4 – 23x2 + 6x+ 2

= 16x6y8z4 = 4y4z2

4x6y4z2

3 4 2

3 2

4

2

x y z

x y z

2

Order of Operations

More Practice

1) −9 − 6(−v + 5) 2) −10(−8x + 9) − 8x 3) 1 + 4(2 − 3k)4) −8v + 6(10 + 6v) 5) 7(1 + 9v) − 8(−5v − 6) 6) −10(x − 7) − 7(x + 2) 7) −2(−6x − 9) − 4(x + 9) 8) 9(7k + 8) + 3(k − 10)

(7x + 2)(5x+1) 5x 17x2

= 35x2+7x+10x +2 = 35x2+17x +2

Multiplying Polynomials

7x *5x 7x *1

2 *5x 2 * 1

Guided Practice

FOIL Method

Column Form

Product of 4x3-32x2+0x +36 * 4Product of 3x4-24x3+0x +27x * 3xSum of the product

Special Products

Square of a Difference (a-b)2 = (a-b)(a-b) = a2-2ab + b2

Find (r- 6)2.

Difference of Squares (a + b)(a – b) = (a-b)(a + b) = (a2 – b2)

Find (m -2n)(m + 2n).

Dividing.

1.

2. =

¿¿¿

(8 𝑥7 𝑦6 𝑧5 )2 𝑥2𝑦 3𝑧 9

Long Division

Individual Practice

Guided Practice

Summary

Homework