Polynomials

Multiplying Monomials

Monomial-a number, a variable, or the product of a number and one or more variables.(Cannot have negative exponent)

› Example: Not Example:

› -5, ½ , 8 5 + a

› 3a, a/2 (½ a) 2a/b

› A2b3 a + b - 6

Constants-monomials that are real numbers

› A number by itself, without a variable (Ex: 4)

When looking at the expression 103, 10 is called the

baseand 3 is called the

exponent or power.103 means 10 • 10 • 10

103 = 1000

An algebraic expression contains:

1) one or more numbers or variables, and

2) one or more arithmetic operations.

Examples:x - 3

3 • 2n41

m

In expressions, there are many different ways to write multiplication.

1) ab 2) a • b 3) a(b) or (a)b 4) (a)(b) 5) a x b

We are not going to use the multiplication symbol any more.

Why?

Division, on the other hand, is written as:

1)

2) x ÷ 3

x

3

Multiplying Monomials

To MULTIPLY powers that have the SAME BASE, just simply ADD the exponents and leave the base the same.

Example:

23 * 25 = 28

x5 * x = x6 (x is the same as x1and 5 + 1 = 6)

Simplify

1. (-7c3d4) (4cd3) = -28c4d7

2.(5a2b3c4) (6a3b4c2) = 30a5b7c6

Find the Power of a Power

To find the power of a power, multiply the exponents.

Example:

(22)3 = 26

Simplify

1.(p3)5 = p15

2.[(32)4]2 = 316

Power of a Product-To find the power of a product, find the power of each factor and multiply.

(a b )m = am bm

EXAMPLE: (-2xy)3 = (-2)3 x3 y3 = -8x3 y3

SIMPLIFY the following:

1). (4ab)2 2). (3y5 z)2

3). [(5cd3)2]3 4). (x + x)2

5). (x3∙x4)3

SIMPLIFYING MONOMIAL EXPRESSIONS

To simplify an expression involving monomials, write an expression in which:

1. Each base appears exactly once. 2. There are no powers of powers. 3. All fractions are in simplest form. SIMPLIFY (⅓xy4 )2 [(-6y)²]³ →(Remember: Start within

your parentheses and

work your way out)

Dividing Monomials

Dividing Powers with the Same Base

To DIVIDE powers that have the SAME BASE, SUBTRACT the exponents.

Quotient of powers: For all integers m and n and any nonzero number a , am = am-n .

an Example: Simplify a⁴ b⁷ = a4-1 b7-2 = a³ b⁵ a b²

Power of a Quotient - For any integer m and any real numbers a and b ,

b ≠ 0, ( a / b )m = am / bm .

Simplify [ 2a³b⁵ ]3 = (2a³b⁵)³

3b2 ( 3b²)³

= 8 a9 b15

27 b6

= 8 a9 b9

27

Power of Zero and Negative Exponents

Zero Exponent : For any nonzero number a , a0 = 1. Example: 30 = 1 , x0 = 1

Negative Exponent Property : For any nonzero number a and any integer n, a-n = 1 and 1 = an .

an a-n

Example: 4-2 = 1

42

Example: 1 = 53

5-3

→The simplified form of an expression containing negative exponents must contain only positive exponents.

1. 313 / 319

Answer: 3-6 = 1 / 36

2. (y³z9) / (yz²)

Answer: y2z7

3. (30h-2 k14 ) / (5hk-3 )

Answer: 6k17

h3

1. b-4 2. (-x-1 y)0 b-5 4w-1 y2

3. (6a-1 b)2 4. s-3 t-5 (b2 )4 (s2 t3 )-1

5. (2a-2 b)-3 5a2 b4

Stacey has to pick an outfit. She has 6 dresses, 12 necklaces, and 10 pairs of earrings. How many different outfits can she choose from if she wears 1 dress, 3 necklaces, and a pair of earrings?

Polynomials

Polynomials

A polynomial is a monomial or a sum of monomials.

Types of polynomials

Binomial: sum or difference of two monomials

Trinomial: sum or difference of three monomials.

Degrees

Degree of a monomial-the sum of the exponents

Example: the degree of 8y4 is 4, the degree of 2xy2z3 is 6 (because if you add all the exponents of the variables you get 6)

Degrees

Degree of a polynomial-the greatest degree of any term in the polynomial Find the degree of each term, the highest is the degree of the

polynomial

Example: 4x2y2 + 3x2 + 5

Find the degree of each term

4x2y2 has a degree 4

3x2 has a degree of 2

5 has no degree

The greatest is 4, so that’s the degree of the polynomial.

Arrange Polynomials

Arrange Polynomials in ascending or descending order

Ascending-least to greatest

Descending-greatest to least

Example: 6x3 –12 + 5x in descending order.

6x3 + 5x –12

Adding and Subtracting Polynomials

When adding or subtracting polynomials remember to combine LIKE TERMS.

Example:

(3x2 – 4x + 8) + (2x – 7x2 – 5)

Notice which terms are alike…combine these terms. (They have been color coded)

3x2 – 7x2 = -4x2

– 4x + 2x = -2x

8 – 5 = 3

So the answer is… -4x2 - 2x + 3

Be sure to put the powers in descending order.

1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)

Group your like terms.9y - 3y - 7x + 8x + 15a - 8a

6y + x + 7a

2. Add the following polynomials: (3a2 + 3ab - b2) + (4ab + 6b2)

Combine your like terms.3a2 + 3ab + 4ab - b2 + 6b2

3a2 + 7ab + 5b2

3. Add the following polynomials using column form:(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)

Line up your like terms. 4x2 - 2xy + 3y2

+-3x2 - xy + 2y2

_________________________

x2 - 3xy + 5y2

4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)Rewrite subtraction as adding the

opposite.

(9y - 7x + 15a) + (+ 3y - 8x + 8a)

Group the like terms.

9y + 3y - 7x - 8x + 15a + 8a

12y - 15x + 23a

5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)Rewrite subtraction as

adding the opposite.(7a - 10b) + (- 3a - 4b)Group the like terms.

7a - 3a - 10b - 4b4a - 14b

6. Subtract the following polynomials using column form:(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)

Line up your like terms and add the opposite.

4x2 - 2xy + 3y2

+ (+ 3x2 + xy - 2y2)--------------------------------------

7x2 - xy + y2

Add or Subtract Polynomials

1. (5y2 – 3y + 8) + (4y2 – 9)

Answer: 9y2 –3y –1

2. (3ax2 – 5x – 3a) – (6a – 8a2x + 4x)

Answer: 3ax2 – 9x – 9a + 8a2x

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b2. 3a + 3b3. 7a + 3b4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b2. 3a + 3b3. 7a + 3b4. 7a – 9b

Multiplying Polynomials

Multiplying a Polynomial by a Monomial

Examples

1. -2x2(3x2 – 7x + 10)

Notice the –2x2 on the outside of the parenthesis……you must distribute this.

-2x2 * 3x2 = -6x4

-2x2 * -7x = 14x3

-2x2 * 10 = -20x2

Answer: -6x4 + 14x3 – 20x2

Examples2. 4(3d2 + 5d) – d(d2 –7d + 12)

Notice you have to distribute the 4 and –d

4 * 3d2 = 12d2

4 * 5d = 20d

-d * d2 = -d3

-d * -7d = 7d2

-d * 12 = -12d

Put it all together….

12d2 + 20d –d3 + 7d2 – 12d

Notice the like terms….

Answer: -d3 + 19d2 + 8d

Multiplying Two Binomials

Example:

(x + 3) (x + 2)

This can be done a number of ways.

Use either FOIL or Box Method

FOIL

(x + 3) (x + 2)

F-Multiply the First terms in each

x * x = x2

O-Multiply the Outer terms

x * 2 = 2x

I-Multiply the Inner terms

3 * x = 3x

L-Multiply the Last terms

3 * 2 = 6

Answer: x2 + 5x + 6

Box Method

Combine

Add the two that are circled

Answer:

x2 + 5x + 6

Polynomials(4x + 9) (2x2 – 5x + 3)

Multiply 4x by (2x2 –5x + 3)

4x * 2x2 = 8x3

4x * -5x = -20x2

4x * 3 = 12x

Multiply 9 by (2x2 –5x + 3)

9 * 2x2 = 18x2

9 * -5x = -45x

9 * 3 = 27

Put it all Together

8x3 – 20x2 + 12x + 18x2 –45x + 27

Now combine like terms

Answer:

8x3 –2x2 –33x + 27

Special Products

A. Square of a Sum: The square of a + b is the

square of a plus twice the product of a and b plus

the square of b.

Symbols: (a + b)² = (a + b)(a + b)

= a² + 2ab + b²

Example: (x + 7)² = x² + 2(x)(7) + 7²

= x² + 14x + 49

Find each product:

1). (4y + 5)² 2). (8c + 3d)²

B. Square of Difference: The square of a – b is the

square of a minus twice the product of a and b

plus the square of b.

Symbols: (a – b)² = (a – b)(a – b)

= a² - 2ab + b²

Example: (x – 4)² = x² - 2(x)(4) + 4²

= x² - 8x + 16

Find each product:

1). (6p – 1)² 2). (5m³ - 2n)²

C. Product of a sum and a difference: The product

of a + b and a – b is the square of a minus the

square of b.

Symbols: (a + b)(a – b) = (a – b)(a + b)

= a² - b²

Example: (x + 9)(x – 9) = x² - 9²

= x² - 81

Find each product:

1). (3n + 2)(3n – 2) 2). (11v – 8w²)(11v + 8w²)

Summary:

Square of a Sum………(a + b)² = a² + 2ab +b²

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

Product of a Sum and a Difference …………….(a – b)(a + b) = a² - b²

Guided Practice:

1. (a + 6)² 2. (4n – 3)(4n – 3)

3. (8x – 5)(8x + 5) 4. (3a + 7b)(3a – 7b)

5. (x² - 6y)² 6. (9 – p)²

7. (p + 3)(p – 4)(p – 3)(p + 4)

Examples

3. y(y – 12) + y(y + 2) + 25 = 2y(y + 5) – 15

Distribute y, y and 2y

y * y = y2

y * -12 = -12y

y * y = y2

y * 2 = 2y

Don’t forget the +25

2y * y = 2y2

2y * 5 = 10y

Don’t forget the -15

Now you have…….

y2 – 12y + y2 + 2y + 25 = 2y2 + 10y –15

Combine like terms….

2y2 –10y + 25 = 2y2 + 10y – 15

Now you have to solve because you have an equals sign

Answer: y = 2