By Kendal Agbanlog

6.1-Measurement Formulas and Monomials6.2-Multiplying and Dividing Monomials6.3-Adding and Subtracting Polynomials6.4-Multiplying Monomials and Polynomials6.5-Multiplying Polynomials6.6-Factoring Trinomials of the Form x2+bx+c6.7-Factoring Trinomials of the Form ax2+bx+c6.8-Factoring a Difference of Squares6.9-Solving Quadratic Equations6.10-Dividing a Polynomial by a Binomial

Sphere:Surface Area:Volume:

Measurement formulas are mostly made of monomials. Monomials are the product of a coefficient and one or more variables. In order to solve the monomial you must substitute a number for each variable.

24 rSA 3

3

4rV

Cylinder: Surface Area: Volume:

rhrSA 22 2 hrV 2

HeightRadius

Cube Surface Area: Volume:

Radius

26xSA 3xV

Multiplying and Dividing Multiplying and Dividing MonomialsMonomials

The Coefficient is the number in front of the variable. The Constant is the number by itself.

Polynomials

Monomial x

2x

Trinomials 5a+8b+5c lmn+6+hij

Binomials a+7

3a+4b

Multiplying

Add the exponents and Multiply the Coefficients… so the answer would be…

3476 33 yxyx 10109 yxDividing

Subtract the exponents and divide the coefficients… so the answer would be…

83

29

6

30

ba

ba6

65

b

a

Common Mistakes: Remember to multiply or divide the negative or positive sign of the coefficient in. When dividing, remember to put answer to the exponent where the larger exponent used to be.

Degree of a Polynomial-

To find the degree of a polynomial, add together the exponents on the variable for each term. The largest number is the degree of the polynomial.

Ex: 4354 231114 yxxyxyyx

The exponent “4” from the x, plus the exponent “1” from the y would equal 5.

The exponent from x plus the exponent of y would equal 2.

The exponent of x plus the exponent of y would equal 6.

The exponent of x plus the exponent of y would equal 7…

Therefore the degree would be 7 since it’s the highest sum.

Common Mistakes: When finding the degree, if it’s just a variable by itself, don’t forget to add one even if it doesn’t have a one as an exponent.

Adding and Subtracting Polynomials-

To add or subtract polynomials, simply combine like terms.

Ex: 14x4y+11x4y+3xy5+2xy5

5 45 25xy y x These two

are like terms

These two are like terms…

Common Mistakes: Remember to switch the variables for each term when adding or subtracting if they are not in alphabetical order so you won’t forget to add them in the end.

)22(2: yxxEx

Multiply 2x with 2x and multiply 2x with –2y.

xyx 44 2

Multiply. :Ex

Multiply 3 with everything inside the brackets…

)54(3 2 xx 15123 2 xx

Factor to check if you have the right answer.

Ex: 2x(2x-2)-(-2x+4)

Multiply everything in the brackets with -1

2x(2x-2)+2x-4

Multiply 2x with (2x-2)

4x2-4x+2x-4Combine like terms

2x2-2x-4

Ex: Expand:

Step 1:

Multiply the coefficients and the constant terms.

)4)(34( xx123164 2 xxx 12194 2 xx

Common Mistakes: Remember to multiply ALL coefficients and constant terms. Never forget the negative sign, and variables too.

Step 2:

Combine like terms.

Quadratic Term

Linear Term

Constant Term

65 : 2 xxEX

)2)(3( xx

xx 2x x3 32

Ex:

Find the common factor, and remove it first.

20155 2 xx

)43(5 2 xx

Find two integers that have a product of –4 and a sum of 3.

)4)(1(5 xx

Common Mistakes- If there is a negative sign, don’t forget about it and make sure you use it when factoring. Another common mistake would be forgetting to find the common factor first.

First, find two integers that have a product of 6 and a sum of 5.

Ex:

Find two integers with the sum of 18 and a product of 8 and –5.

5188 2 xx

)14)(52( xx

Ex: Factor42210 2 yy

Common Mistakes: Don’t forget to bring out the common factor first.

)2115(2 2 yy)15)(2(2 yy

Bring out the common factor firstFind two integers so that the sum of the product of the inside and outside terms is -11 and they have a product of 5 and 2.

Find the square root of 16. One is negative and one positive. Then, factor.

16: 2 xEx

)4)(4( xx1: 4 xEx

)1)(1)(1(

)1)(1(2

22

xxx

xx

Common Mistakes: When factoring a binomial, don’t forget to remove the common factor first.

Quadratic Equations:

8103

,254

,06

2

2

2

xx

x

xx753: 2 xEx25375

5. positive and negativeboth xTherefore,

525

252

x

2.-or 3either equal x wouldTherefore,

2 3

02 03

0)3)(2(

first.Factor

06: 2

xx

xx

xx

xxEx

Common Mistakes- Don’t forget to bring out the common factor first if you can.

Dividing a polynomial by a binomial is just like long division. It has a divisor, quotient, remainder and dividend just like long division.

Ex:____3

63

118322

2

x

xx

xxx

You multiply 3x with x to get 3x2, or you can take 3x2 and divide it by x to get 3x. Then you multiply 3x by 2 to get 6x.

112 xBring down the 11.Then subtract 6x from 8x to get 2x. Take 2x and divide it by x to get 2

42 x

7

Subtract 4 from 11 to get 7 as a remainder.

2 7R

RemainderRDividend,P

Quotient,Q Divisor,D

. get the To

Use

StatementDivision

RDQP

orD

RQ

D

P

When dividing, always look at the first term.And don’t worry about the second one.

7)23)(2(1183x

:be ouldquestion w for thisStatement Division The2 xxx

area. surface and volume theFind 4cm. of radius a has beachballA 1.cylinder. theof volume theFind 2.

5.5cm

Diameter is 3.8cm

( 2ab2)( 3ab)

(5a3b4 )(2ab)

10ab2

3x 7y 4 x 3y

(6x - 2y) - (-2x - 2y)

7. Simplify 4x(5x 3y 5xy)

8. Factor : 20x 2 35x 70

9. Expand (4x-3)(x-9)

10. Factor : (x 4)(x 4)

11. Factor : x 2 6x 9

12. Factor : x2 17x 72

13. Factor : 24x2 13x 2

14. Factor : 4x2 20x 25

15. Factor x2 225

16. Factor 18x 2 98

17. Solve x 2 6x 15 4x

18. Solve 3a2 7a 10

19. Divide x 2 x 3 5x 2 x 1020. Divide a - 3 a3 19a 24

3. Simplify

4. Simplify

5. Simplify

6. Simplify

1. V 268.08cm3, SA 201.06cm2

2.V 62.4cm3

3. 6a3b3

4. a2b3

5. x 4 y

6. 8x

5)-5)(2x-(2x 14.

2)-1)(3x(8x 13.

9)8)(x(x .12

)3)(3( .11

168 .10

27394 .9

)1475(4x .8

201220 .7

2

2

2

22

xx

xx

xx

x

yxxyx 54 103 .20

24 73 .193

10,1 .18

5,3 .17

)73)(73(2 .16

)15)(15( .15

2

2

Raa

Rxx

a

x

xx

xx

1. A beachball has a radius of 4cm. Find the surface area.24 rSA

56637061.124 162 r

206.2011656637061.12 cm

cylinder. theof volume theFind 2.

Diameter is 3.8cm

5.5cm

hrV 2 28.3 r

9.1r

61.39.1 2 34114948.1161.3

34.625.534114948.11 cm

)3)(2.(3 2 abab Multiply the coefficients

6

Add the exponents of like terms

2a 3b

2

43

10ab

)2)((5a4.

abb

Multiply coefficients and add the exponents of like terms

2

54

10

10

ab

baSubtract the exponents of like terms and divide the coefficients

a3b3

yxyx 3473 .5

Change the sign of 2x and 2y.

yx 4

2y)-(-2x-2y)-(6x .6

x8

Combine like terms

2y2x2y-6x Then, combine like terms

703520 : .8 2 xxFactor

)535(4 .7 xyyxx

)1475(4x 2 x

yxxyx 22 201220

Multiply 4x with everything in the brackets

Find a common factor

1682 xx

27394 2 xx

)9)(34( .9 x-x-

10. Expand : (x 4)(x 4)

xxx 844

xxx 39363

7217 x:Factor .12 2 x

96 :Factor .11 2 xx Find two integers with the product of 9 and sum of 6.

)3)(3( xx

xxx 633

9)8)(x(x

Find 2 integers with the product of 72 and the sum of 17.

xxx 1789

25204x :Factor .14 2 x

21324x :Factor .13 2 x

5)-5)(2x-(2x

2)-1)(3x(8x

xxx 13316

xxx 20)10(10

9818 .16 2 x

225 x.15 2

)15)(15( xx

)73)(73(2 xx

Find the square root of 255

First, find a common factor. It would be 2)499(2 2 x

Find the square root of both numbers

1073 .18 2 aa

xxx 4156 .17 2

5,3 x

3

10,1 a

Bring everything to one side

01522 xx

Factor…)3)(5( xxThen solve…

05 x03 xTherefore…

Bring everything to one side01073 2 aaFactor…)1)(103( aaThen solve…

3

10

3

3

103

0103

a

a

a

1

01

a

a

2419a3-a .20 3 a

1052 .19 23 xxxx

54 1032 Raa

24 732 Rxx

Find a number to multiply by x-2 to get x3-5x2

x2 Multiply x2 with x-2

x3-2x2Subtract x3-2x2 from x3-2x2 and bring down the –x.

-3x2-xFind another number and multiply again

-3x2+6x

-3x

Subtract and bring down the -10

-7x-10Again, find a number that goes into -7x-10 and then multiply again

-7

Subtract and find to remainder-7x+14

-24

R-24

First add a 0a2 in between the a3 and –19a

24190a3-a 23 aa

Find a number that goes into a3+ 0a2. It would be a2 a2 Multiply, subtract, and bring down the 19a

a3-3a2

3a2-19a

Find a number that goes into 3a2-19a+3a

3a2-9aSubtract and bring down the –24.

-10a-24Find a number that goes into –28a-24, multiply, subtract and get your remainder.

-10

-10a+30-54

R-54