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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