Monomials and Polynomials

Exponent Rules am · an = am+n x2 · x3 = x5

am = am-n x7 = x2 an x5

a-n = 1 x-3 = 1 an x3

(am)n = amn (x4)2 = x8

(ab)m = ambm (x2y5)3 = x6y15

a0 = 1 x0 = 1

Adding and Subtracting

Combine like terms 3x – 4x = -x

6x2 – 4x + 10x2 – 2x = 16x2 – 6x

(2x + 4) + (3x + 5) = 5x + 9

(3x – 2y) – (6x – 5y) = -3x + 3y

(5x2 + 2x) – (4x + 3) = 5x2 – 2x - 3

Multiplying

Distributing 2x(x + 4) = 2x2 + 8x

-3x3(4x2 – 2x – 6) = -12x5 + 6x4 + 18x3

FOIL – first, outside, inside, last (x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x +6

(x – 4)(x + 4) = x2 + 4x – 4x – 16 = x2 – 16

(x – 5)2 = (x – 5)(x – 5) = x2– 5x –5x + 25 = x2–10x + 25

Dividing

2 3 2 2 3 228x y - 4x y 8x y 4x y

= - = 4x - 2x y2xy 2xy 2xy

2 5 3 2 5 33 2

2 2 2

3x y - 9x y 3x y 9x y = - = 1 - 2x y

3x y 3x y 3x y

Dividing Synthetic Division

Look at the binomial and write the opposite constant in a box.

Look at the polynomial and check for missing terms. If there are any missing terms, hold their space with a zero. Write the coefficients and constant in a row next to the box.

Drop the first number down.

Multiply this number by the number in the box.

Add down.

Multiply this number by the number in the box.

Add down.

Continue this pattern.

(x3 + 2x – 5) (x – 2)

2 1 0 2 -5

2 4 12

1 2 6 7

x2 x # R

Solution: x2 + 2x + 6 +

7x 2

GCF Factoring

GCF – greatest common factor – What is the biggest term that goes into each term? 4 and 6 – GCF = 2 2x2 and 4x – GCF = 2x

GCF Factoring – 1st find the GCF and put it outside the parentheses. 2nd put what is leftover after you divide each term by the GCF inside the parentheses. 3rd check to see if what is inside the parentheses can factor again. 2x2 + 4x = 2x(x + 2) 6x3 + 12x2 – 3x = 3x(2x2 + 4x -1)

Two Terms (Squares) Factoring

Difference of Squares – two squared terms separated by a subtraction sign.

To factor these you need two parentheses. Take the square root of the 1st term and place this 1st in each parentheses. Take the square root of the 2nd term and place this 2nd in each parentheses. One set of parentheses gets a plus and the other gets a minus.

x2 – 9 = (x + 3)(x – 3) 4x2 – 49 = (2x + 7)(2x – 7)

Two Terms (Cubes) Factoring

There are two terms that are cubes and are separated by a plus or a minus.

(x3 – 27) = (x – 3)(x2 + 3x + 9)

(x3 + 8) = (x + 2)(x2 – 2x + 4)

Four Terms

Grouping Group the first two terms together and group the

last two terms together. Look at the first group and factor out the GCF. Look at the second group and factor out the GCF. Check to make sure both parentheses are the

same. Group the GCFs in one set of parentheses and

group the common factors in the other parentheses.

Three Terms Factoring The first term is only an x2

Look at the last term and list each of its factors. Look at the last sign.

If it is a plus, pick the two factors that add to give you the middle term. If the middle sign is also a plus, then your answer will look

like this ( + )( + ) If the middle sign is a minus, then your answer will look like

this ( - )( - ) If it is a minus, pick the two factors that subtract to give you

the middle term. If the middle sign is a minus, the bigger factor gets a minus

sign next to it. If the middle sign is a plus, the bigger factor gets a plus

sign next to it.

Three Terms Factoring The first term has a coefficient other than 1. Multiply the first and last term together. List the factors of this. Look at the last sign.

If it is a plus, pick the two factors that add to give you the middle term. If the middle sign is also a plus, then your factors should both be

positive. If the middle sign is a minus, then your factors should both be

negative. If it is a minus, pick the two factors that subtract to give you the

middle term. If the middle sign is a minus, the bigger factor gets a minus sign next

to it. If the middle sign is a plus, the bigger factor gets a plus sign next to it.

Replace the middle term with the factors from the last step. Do 4 term grouping to get your final answer.