Ch. 12-3 to 12-7

Simplifying Polynomials

Vocabulary• Monomial is:

–A number, (e.g. -5, 0, 7, …)

–A variable, (e.g. a, b, c, ….)

– or a product of numbers and/ or variables. (e.g. 5*10, 3abc, xyz )

• Polynomial is an algebraic expression that is the sum or difference of one or more monomials. (e.g. -5a + 7b – 10c + 6a - 5b)

The like terms in this expression are 3r and –r.

Write the polynomial.

= 3r – r + 8p – 6q Group like terms

Simplify by combining like terms.

Answer:

Answer:

There are no like terms in the expression.

Answer:

Answer: simplest form

Rewrite the polynomial from the greatest powers to the least. Then group and add like terms.

Answer:

= 8x² - 2x² + 2x – 9x + 3

2x + 8x² – 9x + 3 – 2x²

= (8 – 2) x² + (2 – 9)x + 3

= 6x² + – 7x + 3

Answer:

Ch. 12-4

Adding Polynomials

Method 1 Add vertically.

Method 2 Add horizontally.

Answer: The sum is

Align like terms.

Add.

Associative and Commutative Properties

Answer:

Ch. 12-5

Subtracting Polynomials

= 8c + 3 - + 6c - + 2 - + = -

= 8c + 3 – 6c - 2

= 8c – 6c + 3 – 2 Combining like terms= (8 - 6)c + (3 – 2)

= 2c + 1

The answer is 2c +1

Answer:

= 6z + 1 – 2z + 5 - + = -; - - = +

= 6z – 2z + 1 + 5 Combining like terms

(6z + 1) – (2z – 5) = 6z + 1 – + 2z - - 5 Remove parenthesis

= (6 – 2)z + (1 + 5)

= 4z + 6)

Answer: The difference is 4z + 6

Answer:

(10f² - 15) – (- 5f + 3) = 10f² - 15 - - 5f - + 3

= 10f² - 15 + 5f – 3 - - = +, - + = -

= 10f² + 5f – 15 – 3 arrange from greatest power to the least

= 10f² + 5f -18

Answer: The difference is 10f² + 5f -18

Answer:

Ch. 12-6

Multiplying and Dividing Monomials

Important Concepts: Product of Powers

3² * 3³

3² = 3 * 3

3³ = 3 * 3 * 3

So, 3² * 3³ = 3 * 3 * 3 * 3 * 3 = 35

We found out that when multiplying two exponents with the same base, we can simply add their powers.

3² * 3³ = 32+3 = 35

In the same case, r³ * r² = r2+3 = r5

The common base is 7.

Add the exponents.

Check

Answer:

Find . Express using exponents.

Find . Express using exponents.

Answer:

Find . Express using exponents.

Answer:

Commutative and Associative Properties

The common base is x.

Add the exponents.

Find . Express using exponents.

Answer:

Important concepts: Quotient of powers

5³ 5*5*5

5² 5*5

We can also do 5³‾² = 5¹ = 5

In the same way,

r

r³

This is called Quotient of Powers

= = 5

5

= r ‾³ = r²5

The common base is 6.

Simplify.

Answer:

Find . Express using exponents.

Answer:

Find . Express using exponents.

The common base is a.

Simplify.

Answer:

Find . Express using exponents.

Answer:

Find . Express using exponents.

UNIT CONVERSION One centimeter is 10 millimeters, and one kilometer is 106 millimeters. How many centimeters are there in one kilometer?

To find how many centimeters there are in one kilometer, divide 106 by 10.

Quotient of Powers

Simplify.

Answer: There are 105 centimeters in one kilometer.

Answer: 104 decimeters

UNIT CONVERSION One decimeter is 10 centimeters, and one kilometer is 105 centimeters. How many decimeters are there in one kilometer?

Distributive Property

Answer:

Answer:

Distributive Property

Definition of subtraction

Answer:

Answer:

Distributive Property

Answer:

Answer:

Distributive Property

Simplify.

Definition of subtraction

Answer:

Answer:

Power Rule:(3²)³ = 9³ = 9 * 9 * 9 = 729

(3²)³ = (3²) (3²) (3²) = 9 * 9 * 9 = 729

3²*³ = 3 = 3*3 * 3*3 * 3*3 = 729

We found out that when there is another power over an exponent, we can find the value by multiplying their powers. Same rule applies to algebraic expressions

(x³)³ = x³*³ = x

6

9

Example 1: Power of a product

(-2xy)³

= -2¹*³x¹*³y¹*³ any # to the 1st power is the

number itself

= (-2)(-2)(-2) x³y³ -2³ = (-2)(-2)(-2) = -8

=-8x³y³

Answer: -8x³y³

Your turn:

(-3ab)²

= -3¹*²a¹*²b¹*² any # to the 1st power is the

number itself

= (-3)(-3) a²b² -3² = (-3)(-3) = 9

= 9a²b²

Answer: 9a²b²

Example 2: Power of a power

(2²xy³)²

= 2²*²x¹*²y³*² any # to the 1st power is the

number itself

= (2)(2)(2)(2) x²y

=16x²y

Answer: 4x²y

6

6

6

Your turn:

(-3²a³b)²

= -3²*²a³*²b¹*² any # to the 1st power is the

number itself

= -3 a b²

= 81a b²

Answer: 81a b²

6

4

6

6

Example 3: Product rule and Power rule

(4cd)² (-3d²)³

= (4¹*²c¹*²d¹*²) (-3¹*³d²*³) Follow the power rule

= (4²c²d²) (-3³d )

= (16c²d²) (-27d )

= (16)(-27)(c²)(d²)(d )

=-432c²d² Product rule

=-432c²d

6

6

6

+ 6

8

Your turn:

(-2x )³ (-5xy )²

=-200x y

5 6

17 12