J. Baker February 2010 1

J. Baker February 2010 2

• Monomials: a number, variable, or the product of quotient of a number and variable.

• Polynomial: a monomial or the sum of 2 or more monomials– Types of Polynomials

• Monomials (1 term)• Binomials (2 terms)• Trinomials (3 terms)

BEWARE: a quotient with a variable in the denominator is NOT a polynomial!

Example: A or 2 B x2

J. Baker February 2010 3

ExamplesMonomials NOT monomials

12 A + b

X 5 – 7d + 2f

-4x2

11ab1/3 xyz2

5

w2

Classify the following in a table on the next page:

4x – 7-9X2 +2xy +y2

14m2abc2 + 13x 5

3x2 -11xyA + 2b + 4c2x + 9y3y2 3a – 7b2 -4c2x2 + 5x + 4

J. Baker February 2010 4

Classify by number of terms Monomial Binomial Trinomial

Degree of a monomial: The sum of the exponents of its variables

8y3 3

4y2 a1b1 4

-14 0

42a1b1c1 3

+

Hint: variableBy themselvesHave a degreeOf 1

J. Baker February 2010 5

Degree of a polynomial with more than one term:The GREATEST degree of all terms

3x2 + 8a2b + 4 2, 3, 0 3

7x4 + 4x - 9x2y7 4, 1, 9 9

Polynomial degree of terms degree of polynomial

J. Baker February 2010 6

Ordering Polynomials: the terms are written in ascending or descending order with respect to one variable’s exponents

Ascending: exponents of the specified variable go from smallest to largest (up)

Descending: exponents of specified variable go from largest to smallest (down)

Example: 1.

2.

Example:

1.

2.

J. Baker February 2010 7

Adding polynomials: add the coefficients of like terms and keep the variables and exponents

the same Parenthesis are not necessary in this step, the simply separate terms to be added

1, (4x + 6y) + (3x + 9y) = 7x + 15y

2. (3x2 – 5xy + 8y2) + (2x2 + xy – 6y2)

= 5x2 – 4xy + 2y2

3. (3p2 – 2p + 3) = (p2 + 7) = 4p2 – 2p + 10

J. Baker February 2010 8

Subtracting Polynomials: * Distribute the negative to everything behind it* change the “ – “ to “ + “ and add as before

(7x2 – 8) – (-3x2 +1) = 7x2 – 8 + 3x2 -1 = 10x2 – 9

(x2 + y2) – (-x2 + y2)= x2 + y2 + x2 – y2

= 2x2

(2a2 – ab + b2) – (3a2 + 5ab – 7ab2)

= 2a2 – ab + b2 – 3a2 – 5ab + 7ab2

= -a2 -6ab + b2 + 7ab2

J. Baker February 2010 9

Finding Perimeter: add all sides(add coefficients and leave variables the same)

Square

-3x2

-3x2 + -3x2 + -3x2 + -3x2

= -12x2

2a

-6b2

-3a + b2

2a + -6b2 + -3a + b2

= -a -5b2

-m + 5n

2m – 4n

2m - 4n + -m + 5n + 2m - 4n + -m + 5n

= 2m + 2n

J. Baker February 2010 10

Laws of Exponents

Review: monomial = 1 term (no + or - )

2x5

coefficient variable

exponent

Product of Power: “multiplying with exponents”

General rule: am * an = am+n 1. b3 * b8 = b11

2. x (x3) (x4) = x8

3. (a2b) (a5b4) = a7b5

4. (6c2d3) (5c4d) = 30c6d4

5. (10w2) (-7w2y3) = -70w4y3

6. (a2b) (3a5b6) = 3a7b7

7. (-ab2) (5a2) (-b3) = -5a3-b5

Add the exponents

J. Baker February 2010 11

Power of a Power

General rule: (am)n = amn [multiply the exponents]

1. (x5)2 = x10

2. (a2)6 = a12

3. (b)8 = b8

4. (c5)4 = c20

Power of a Product

General Rule: (ab)m = ambm

(backwards distributing)

1. (ab)4 = a4b4

2. (c2d3)5 = c10d15

3. (-2x3)4 = 16x12

4. (-9ax3y2)3 = -729a3x9y6

J. Baker February 2010 12

Mix it Up! Use all your rules!

1. (2d)2(5d3)

2. -2x(6x2)3

3. (-3ab)3(2b3)

4. (3x)2(4x2yz6)

5. (3a)(-a2b)2(4ab2c)

6. -5(2p2)3

7. (-3ab)(-3ab3)(-3a2b4)3

J. Baker February 2010 13

Quotient of Powers: dividing with exponents

General rule: am = am-n

an

Subtract the exponents

X5

X2

4x9

x3

m4w3

mw2

10m4

30m

50x5y3

-25x2y= X5-2 = X3

= 4x9-3 = 4x6

= m4-1 w3-2 = m3w

= 10 m4-1 = 1 m3

30 3

= 50 x5-2 y3-1

-25= -2x3y2

Reminder: variables that stand alone

have an exponent of 1

J. Baker February 2010 14

Zero ExponentB4

B4 = B4-4 = B0 = 1 Think of it as 5 or -10 or B4

5 -10 B4

General Rule: a0 = 1

ANYTHING to the zero power = 1

Negative ExponentN3

N7 = n-4 ***cannot have negative exponents***

To remove a negative exponent move only the number or variable up or down

General Rule: a-n = 1 or 1 = bn

an b-n

J. Baker February 2010 15

Mix it up!

1. 3x-2

2. -5a2b-3

3. 5a-1

10b-2

4. 15a-3

45a-2

5. (6a-1b)2

(b2)4

6. 24w3t4

6w7t2

7. 15x3

5x0

J. Baker February 2010 16

Distribute monomial into polynomial– Multiply coefficients– Add exponents

Multiplying Monomial x Polynomial

3(2x - 5) = 3*2x – 3*5= 6x -15

7b( 4b2 – 18)

7b( 4b2 – 18)= (7*4 b1+2) – (7b * 18)= 28b3 – 126b

3d(4d2-8d-15)

3d(4d2-8d-15)= (3*4 d1+2) – (3*8 d1+1) – (3*15 d1)= 12d3 – 24d2 – 45d

J. Baker February 2010 17

Binomial * Binomial FOIL

F first: multiply the first monomials in each set of ()

A*CO outside: multiply monomials

on the outside of each set ().

A*DI inside: multiply monomials on

the inside of each set ().

B*CL last: multiply monomials in the

last position in each set ().

B*D

(2x * 4x) + (2x * 4) + (3 * 4x) + (3 * 4)

8x2 + 8x + 12x + 12

8x2 + 20x + 12

(A + B)(C + D)

(A*C) + (A*D) + (B*C) +(B*D)

(2x +3)(4x +4)

Combine like terms if possible

J. Baker February 2010 18

Binomial * Binomial Box

Place binomials on either side of the box and multiply each box (Similar to Punnett squares in biology)

8x2

1

12x2

8x3

124

(2x + 3) (4x + 4)

2x +3

4x

+4

Combine like terms (boxes 2 and 3)

8x2 + 8x + 12x + 12

8x2 + 20x + 12

J. Baker February 2010 19

Polynomials: rules to remember

COEFFICIENTS(number in front of variable)

EXPONENTS

ADDing Add coefficients of like terms

Leave exponents and variables alone. They are

used only for combining like terms in this step

SUBTRACTing-distribute the negative

-add

First set of ( ) never change

Distribute the negative to everything behind it

Change “-” into “+” and add as normal

Leave exponents and variables alone. They are

used only for combining like terms in this step

MULTIPLYing Multiply numbers as normal Add exponents of like terms

DIVIDing Divide numbers as normal Subtract exponents of like terms. No Negative exponents allowed

Exponents(outside parentheses)

Raise coefficient to the power shown outside

parenthesis

Multiply exponent inside parenthesis with the exponent outside the

parenthesis