unit 1, part 1 linear functions - weebly
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3
Expressions/Polynomials โ Parts, Standard Form, Classifying
Terms: _______________________________________________________________________
Variables: ___________________________________________________________________
Coefficients: _________________________________________________________________
Constants: ___________________________________________________________________
Like Terms: ___________________________________________________________________
Degree: _____________________________________________________________________
Examples:
1) 6๐ฅ โ 2๐ฆ + ๐ฅ โ 5 2) 2๐ + ๐ + 8๐ โ 1
Terms: _____________________________ Terms: _____________________________
Like Terms: _________________________ Like Terms: _________________________
Variables: __________________________ Variables: _________________________
Coefficients: _______________________ Coefficients: _______________________
Constants: _________________________ Constants: _________________________
Standard Form of a Polynomial: ____________________________________________
___________________________________________________________________________
Classifying Polynomials (Naming Polynomials)
By Degree By Number of Terms
Degree Name # of Terms Name
0 1
1 2
2 3
3 4 or more
4
5
6 +
4
Identifying Parts of Algebraic Expressions Practice
1) How many terms are in each of the following algebraic
expressions?
Answers:
a) 6๐ฅ3 + 8๐ฅ2 โ 4๐ฅ
b) 15๐ฅ๐ฆ3 + 21๐ฅ2 โ 16
c) 19๐ฅ4 + 8๐ฅ2 + 4๐ฅ๐ฆ โ 2
d) 8๐ฅ3 + 14๐ฅ5 โ 20๐ฅ2 + 9๐ฅ โ 25
e) 9๐ฅ3๐ฆ + 5๐ฅ4 โ 24๐ฅ2 + 7๐ฅ โ 6๐ฅ6
f) 2๐๐ + 7
g) 15๐ฅ๐ฆ + 7๐ฅ + 2๐ฆ + 9
2) Identify the coefficients, constants,
and variables in each expression. Coefficients Constants Variables
a) 81๐ฅ3 + 7๐ฅ๐ฆ2 โ 14๐ฅ
b) 4๐ฅ3 + 8๐ฅ2 โ 24
c) 61๐ฅ2 + 6๐ฅ + 7
d) 4๐ฅ๐ฆ๐ง3 + 8๐ฅ2 โ 2๐ฅ๐ฆ2 + 29๐ฅ โ 46
e) 22๐3 + 38๐2 โ 12๐
f) 28๐2 โ 17๐๐
g) 7๐ฅ + 2๐ฅ๐ฆ
3) Identify the exponents in each expression. Answers:
a) 12๐ฅ3๐ฆ2
b) 62๐ฅ4
c) 2๐ฅ2๐ฆ
d) 125๐ฅ5
e) 9๐7
f) โ12
g) โ12๐๐2๐
4) List the like terms in each of the following algebraic
expressions?
Answers:
a) 14๐ฅ๐ฆ2 + 25๐ฅ โ 6๐ฅ + 2
b) 8๐ฅ2 + 12๐ฅ2 โ 9๐ฅ๐ฆ + 3๐ฅ
c) 86๐ฅ3 + 42๐ฅ โ 36๐ฅ3 + 21๐ฆ
d) 4๐ฅ2 + 6๐ฆ โ 6๐ฅ + 7๐ฆ
e) 36๐3 + 22๐2๐2 โ 2๐2๐2 + 7๐ โ 50
5
Standard Form and Classifying Polynomials Practice
Standard Form Classify by
Degree
Classify by #
of Terms
1) โ2๐ฅ2 โ 3๐ฅ5 + 7
2) 7
3) 6๐ฅ4
4) 7๐ฅ2 โ 2๐ฅ2
5) โ2 + 4๐ฅ4
6) 5๐ฅ5 + 3๐ฅ3 โ 2 โ 4๐ฅ
7) โ6๐ฅ2 + 9๐ฅ2
8) โ2๐ฅ3
9) 7 โ 8๐ฅ + 4๐ฅ3
10) 7๐ฅ2
11) 5 + 9๐ฅ3 + 3๐ฅ3
12) โ5 + ๐ฅ
13) โ8๐ฅ4 โ 8๐ฅ
14) 3๐ฅ โ 5๐ฅ2 โ 2
15) 3๐ฅ3 + ๐ฅ
6
Adding and Subtracting Polynomials
Adding Polynomials: Subtracting Polynomials:
(4๐ฅ2 + 8๐ฅ โ 9) + (โ2๐ฅ2 + 11) (4๐ฅ2 + 8๐ฅ โ 9) โ (โ2๐ฅ2 + 11)
Let ๐(๐ฆ) = 4๐ฆ3 + 5๐ฆ โ 2, ๐(๐ฆ) = โ5๐ฆ + 12, โ(๐ฆ) = โ๐ฆ2 + 7๐ฆ โ 1, and
๐(๐ฆ) = 4๐ฆ2 + 6๐ฆ + 5.
Find the following:
1) ๐(๐ฆ) + โ(๐ฆ) 2) ๐(๐ฆ) โ ๐(๐ฆ)
3) ๐(๐ฆ) + ๐(๐ฆ) 4) โ(๐ฆ) + ๐(๐ฆ)
More Examples:
1) Find the sum of (2๐ฅ3 + 8๐ฅ) and (โ7๐ฅ3 + 3๐ฅ2 + 11๐ฅ).
2) Find the difference of (8๐ฅ3 โ 2๐ฅ2 + 14) and (โ๐ฆ3 + 4).
7
Adding and Subtracting Polynomials Practice
Simplify each expression.
1) (2๐ฅ2 โ 8๐ฅ4) โ (4๐ฅ2 โ 8) 2) (3 โ 3๐4) + (8๐4 + 7๐2)
3) (4๐ฅ4 + 6๐ฅ3) + (6๐ฅ3 + 8๐ฅ4) 4) (6๐4 โ 8) โ (7 โ 5๐4 โ 7๐3)
5) (6๐2 โ 7๐3) โ (๐ โ 7๐3 + 4๐2) 6) (5๐ฅ3 + ๐ฅ) + (๐ฅ โ 3๐ฅ3 โ 3)
7) (6๐ฅ4 โ 4๐ฅ + ๐ฅ2) + (2๐ฅ โ 2๐ฅ2 โ 8๐ฅ4) 8) (5๐4 + 7 โ 6๐) โ (8 + 8๐3 + 2๐4)
9) (8๐ฅ4 โ 8 โ 5๐ฅ) + (5 โ 5๐ฅ3 โ 2๐ฅ4)
10) (8 โ 3๐ฅ + 5๐ฅ3) + (8๐ฅ4 โ 8๐ฅ3 โ 6) โ (8๐ฅ + 3๐ฅ4 โ 5)
11) (4๐4 + 4๐2 โ 5๐) + (2 + ๐2 + ๐) โ (6๐ โ 5๐3 + 8๐4)
8
Multiplying Polynomials
When multiplying polynomials, use the _______________________________________!!
Remember, when multiplying _________________, you ______ the exponents.
๐ฅ โ ๐ฅ = ๐ฅ2 โ ๐ฅ3
Examples:
1) 5(๐ฅ + 6) 2) ๐ฅ2(๐ฅ + 6)
3) (โ2๐ฅ)(๐ฅ2 โ 4๐ฅ + 2) 4) (๐ฅ โ 2)(๐ฅ + 4)
5) (๐ฅ + 9)(๐ฅ โ 3) 6) (๐ฅ + 3)(๐ฅ โ 3)
7) (2๐ฅ + 5)(๐ฅ + 6) 8) (3๐ฅ โ 1)(2๐ฅ โ 4)
9) (5๐ โ 6)(3๐2 โ 2๐ + 5)
10) Find the area of the rectangle 11) Find the volume of the following
below. rectangular prism.
9
Multiplying Polynomials Practice
1) 5๐ฅ3(4๐ฅ2 โ 3๐ฅ + 1) 2) (๐ฅ + 4)(๐ฅ โ 6)
3) (๐ฅ + 9)(๐ฅ โ 9) 4) (3๐ฅ + 1)(2๐ฅ โ 5)
5) (6๐ฅ โ 3)(4๐ฅ โ 1) 6) (8๐ฅ + 7)(2๐ฅ + 3)
7) (2๐ฅ + 5)2 8) (3๐ฅ โ 8)2
9) (๐ฅ + 5)(๐ฅ2 โ 7๐ฅ + 4) 10) (๐ฅ โ 3)(๐ฅ2 + 8๐ฅ + 1)
11) Write an expression for the perimeter and area of the following rectangle.
Perimeter: Area:
10
Solving One-Step Equations Make sure you show your work for each problem.
1) ๐ฆ + 9 = 23 2) ๐ฅ
4= 16
3) 78 +๐ = 100 4) 3๐ = 39
5) ๐ โ 15 = 10 6) ๐ โ 56 = โ10
7) 8๐ = 96 8) ๐
โ7= 3
9) ๐ฅ โ 37 = 25 10) ๐ โ 65 = 185
11) 11๐ก = 143 12) ๐ + 40 = 20
11
Solving Two-Step Equations Make sure you show your work for each problem.
1) 3๐ฆ + 6 = 12 2) 10 + 3๐ฃ = 25
3) ๐
3+ 5 = 9 4) 2๐ฅ + 8 = 16
5) 8 = 5๐ โ 12 6) ๐
10โ 20 = 0
7) 6 =๐โ9
3 8) 4๐ค โ 6 = 10
9) 9๐ โ 40 = 41 10) 14 = 2๐ + 10
12
Solving Multi-Step Equations Make sure you show your work for each problem.
1) 6๐ + 5๐ = โ11 2) 4๐ฅ + 6 + 3 = 17
3) 6๐ โ 1 + 6๐ = 11 4) โ10 = โ14๐ฃ + 12๐ฃ
5) โ6๐ โ 2๐ = 16 6) 0 = โ5๐ โ 2๐
7) ๐ + 11 + 8๐ = 29 8) โ10๐ + 9๐ = 12
9) 37 = โ3 + 5(๐ฅ + 6) 10) โ(๐ โ 8) = โ2
11) 8 = 8๐ฃ โ 4(๐ฃ + 8) 12) 8(1 + 5๐ฅ) + 5 = 13 + 5๐ฅ
13
Solving Inequalities When solving inequalities, follow the same steps that you would when solving
equations. However, there are two things that you need to remember that are
different.
If you _________________________ BOTH sides of the inequality by a
_______________________________, you must ________ the inequality symbol
For your final answer, you want the ___________________ to be on the
________________________ of the inequality
Once you have your answer, you will also need to shade the number line
appropriately.
Less Than Greater Than
Less Than or Equal To Greater Than or Equal To
Make sure you show your work for each problem.
1) 4๐ฅ > 4 2) ๐ โ 20 โฅ โ8
3) 9 โค 6 โ ๐ 4) โ2 > 1 + ๐
14
5) ๐ฅ+8
26โค 1 6) โ44 โค 1 โ 3๐
7) 6 โฅ๐
7+ 8 8) 5(6 + ๐) โค 30
9) ๐ + 3 + 4๐ < โ7 10) 106 โค โ8 + 6(โ4๐ โ 1)
15
Solving Literal Equations
To solve a literal equation:
1) Locate the __________________ that you want to solve for
2) Follow the rule for ____________________________ to get the variable all
by itself (isolated)
Examples:
16
Solving Literal Equations Practice
1) Solve for k. 2) Solve for v.
๐ + 20 = ๐ก ๐ฃ
5= ๐ค
3) Solve for m. 4) Solve for C.
2๐ โ ๐ = 11๐ ๐น =9
5๐ถ + 32
5) Solve for b. 6) Solve for L.
๐ด = ๐โ ๐ = 2๐ฟ + 2๐
For questions 7-10, solve for y.
7) 2๐ฆ = 4๐ฅ + 10 8) 19 = 7๐ฅ + ๐ฆ
9) 8๐ฆ โ 4๐ฅ = 2 10) 3๐ฆ โ 12๐ฅ = 18
17
Translating Words into Expressions, Equations, and Inequalities
Remember, the main difference between an expression and an equation is that
_____________________________________________________________________________.
The difference between an equation and an inequality is that _________________
_____________________________________________________________________________.
Key Words
sum: _________________ difference: _________________
product: _________________ quotient: _________________
less than: _________________ more than: _________________
is: _________________ equals: _________________
twice: _________________ double: _________________
half: _________________ triple: _________________
quadruple: _________________ no more than: _________________
at least: _________________ is less than: _________________
Examples
Translate the following into algebraic expressions, equations, or inequalities.
1) The sum of 8 and t. 2) The difference of 32 and x.
3) Eight more than x. 4) 12 less than some number.
5) Arthur is 8 years younger than Janet. 5) The product of 7 and b is 63.
6) Twice as many points as Bob. 7) Henry is half the age of Sally.
8) Two more than the quotient of 5 and a. 8) Six less than twice a number is 4.
9) The product of 7 and a number is no more than 140.
18
Word Problems
Special Types
Consecutive Numbers
The sum of 4 consecutive integers is 441. Find the four integers.
Consecutive Even/Odd Integers
The sum of three consecutive odd integers is 333. Find the three integers.
Average
Jimmyโs first five unit test grades were 90, 74, 82, 68, and 76. There are a total of
6 unit tests in Jimmyโs science class this semester. Is it possible for Jimmy to have
a test average of 82? If so, what would he need to make on his next test?
Perimeter
A length of a rectangle is 3 inches greater than the width. The rectangle has a
perimeter of 46 inches. Find the length and the width.
19
Word Problems Practice
1) You are trying to save $20 a week to buy a new CD player. During the last 4
weeks you have saved $35, $15, $10, and $12. How much do you need to save
this week in order to average saving $20 per week for the 5 weeks?
2) On an algebra test, the highest grade was 42 points higher than the lowest
test grade. The sum of the two grades (the lowest and the highest grades) was
138. Find the lowest test grade.
3) When 6 is added to four times a number, the result is 50. Find the number.
4) The sum of three consecutive integers is 159. Find the three integers.
20
5) The width of a rectangle is 8 inches more than the length. The perimeter is 32.
Find the length and width of the rectangle.
6) Five times the sum of a number and two is thirty-five. Find the number.
7) Twelve subtracted from three times a number is fifteen. Find the number.
8) Twice a number added to seven is thirteen. Find the number.
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