1. Polynomials and Logarithmic Functions
مدرسة المواهب للتعليم الثانوي
Polynomials and Logarithmic Functions
Grade 11 Academic
2012−2013
Name: …………………………………………………………………………………….....
Class: 11 Ac / ……...

2. Date: ……………………………….
Polynomials
Q1: Why each of the following terms are not a polynomial term?
Q2: Which of the following algebraic expressions are not polynomials? a. b. c. d. e. f.
Q3: Write down three polynomials and find the degree, leading term, leading coefficient and constant term. 1

3. Q5: If a polynomial function is defined by , find: The degree of P(x)
Date: ……………………………….
Polynomials
Q4: For the following polynomials give the degree, the coefficient of the highest power of x and the constant term.
Q5: If a polynomial function is defined by , find:
The degree of P(x)
The coefficient of x2
The constant term
The value of P(1), P(0), P(-1) a. b. c.
Q6: For the following polynomials find the values indicated. 2

4. Adding and Subtracting Polynomials
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Adding and Subtracting Polynomials
Q1: Simplify each expression
5 𝑝 2 −3 +( 2𝑝 2 − 3𝑝 3 )
𝑎 3 − 2𝑎 2 − 3𝑎 2 − 4𝑎 3
4+ 2𝑛 3 +( 5𝑛 3 +2)
4𝑛 − 3𝑛 3 −( 3𝑛 3 +4𝑛)
3𝑎 2 +1 −(4+ 2𝑎 2 )
( 4𝑟 𝑟 4 ) −( 𝑟 4 − 5𝑟 3 )
5𝑛+4 − 5𝑎+3 3

5. Adding and Subtracting Polynomials
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Adding and Subtracting Polynomials
Q1: Simplify each expression
3𝑥 4 −3𝑥 − 3𝑥 − 3𝑥 4
−4𝑘 𝑘 2 +( −3𝑘 4 − 14𝑘 2 −8)
3 − 6𝑛 5 − 8𝑛 4 −( −6𝑛 4 −3𝑛 − 8𝑛 5 )
12 𝑛 5 −6𝑛 −10𝑛 3 − 10𝑛 −2𝑛 5 − 14𝑛 4
8𝑛 −3𝑛 𝑛 2 −( 3𝑛 𝑛 4 −7)
𝑦 3 −7 𝑥 4 𝑦 4 + −10 𝑥 4 𝑦 3 + 6𝑦 3 +4 𝑥 4 𝑦 4 −( 𝑥 4 𝑦 3 +6 𝑥 4 𝑦 4 ) 4

6. Multiplying Polynomials
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Multiplying Polynomials
Q1: Find each product
6𝑣 (2𝑣+3)
7 (−5𝑛−8)
2𝑥(−2𝑥−3)
−4 (𝑦+1)
2𝑛+2 (6𝑛+1)
(4𝑛+1) (2𝑛+6)
𝑥 −3 6𝑥 −2 5

7. Multiplying Polynomials
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Multiplying Polynomials
Q1: Find each product
8𝑝 −2 (6𝑝+2)
6𝑝+8 (5𝑝−8)
3𝑚−1 (8𝑚+7)
2𝑎 −1 (8𝑎−5)
5𝑛+6 (5𝑛−5)
( 6𝑛 2 −6𝑛−5) ( 7𝑛 2 +6𝑛−5) 6

8. Q1: Divide the following polynomials:
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Dividing Polynomials
Q1: Divide the following polynomials:
𝑛 2 −𝑛−29 ÷(𝑛−6)
𝑚 2 −7𝑚−11 ÷ 𝑚 −8
𝑛 2 +10𝑛+18 ÷(𝑛+5)
𝑘 2 −7𝑘+10 ÷ (𝑘−1)
𝑛 2 −3𝑛−21 ÷(𝑛−7) 7

9. Q1: Divide the following polynomials:
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Dividing Polynomials
Q1: Divide the following polynomials:
𝑎 2 −28 ÷(𝑎−5)
𝑟 2 +14𝑟+38 ÷ 𝑟+8
𝑥 2 +5𝑥+3 ÷(𝑥+6)
50𝑘 𝑘 2 −35𝑘−7 ÷ (5𝑘−4) 8

10. *IF a given polynomial p(x) is divided by (x-a) the remainder is p(a)
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Remainder Theorem
*IF a given polynomial p(x) is divided by (x-a) the remainder is p(a)
Q1 : Use the remainder theorem to find the remainder when :
x4 – 3x3 +2x-7x+1 is divided by x-1 . cheek your answer by long division
Q2 : Use the remainder theorem to find the remainder when
the first polynomial is divided by the second:-
(x3-3x2+2x-1) ,(x-1)
(x4-2x3+5x+2),(x-2)
(3x2+7x-4),(x+2)
(x3-3x2-5x+5),(x-4) 9

11. Q3 : Find the remainder when when f(x) = 4x3-3x2-2x-8 is divided by:
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Remainder Theorem
Q3 : Find the remainder when when f(x) = 4x3-3x2-2x-8 is divided by:
(a) x (b) x+1
(c) x (d) x+2 10

12. b) is (x-2) a factor of f(x) ?
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Remainder Theorem
Q4 : a) What is the remainder when f(X) = x3-2x2-x+2 is divided by (x-2) ?
b) is (x-2) a factor of f(x) ?
d) write f(x) as the product of three factors.
Q5 : find k so that when 2x3 –x2-x +k is divided by x-2 , the remainder is zero.
Q6 : find the coefficient m if 3x2 –mx +4 is exactly divisible by x+1 11

13. For any polynomial p(x) :
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The Factor Theorem
*THE FACTOR THEOREM:
For any polynomial p(x) :
(i) if p(a) =0 then (x-a) is a factor of p(x)
(ii) if (x-a)is a factor of p(x) then p(a)=0
Q1 : In each case show that the first polynomial is a factor of p(x) and hence find all the factors of p(x) .
(x-1) , p(x) = x3 + 4x2 + x – 6
(x-3) , p(X)= x4 -3x3 + 3x2 -7x – 6
(x+2) , p(X) = x3+3x2+7x+10
(x+1) , p(X)= 2x3 +9x2 -8x -15
Q2 : find the factor of the following polynomials.
x3 +x2+x +1
x3- 3x2 +3x-1
X3-1
6x3 +7x2-9x +2
Q3 : For what value of k is (x-1) a factor of x2 - 3x + k ?
Q4 : For what value of b is (x+1) a factor of x3 –bx +3 ? 12

14. Date: ……………………………….
The Factor Theorem
Q5: If x3 + ax + b is divisible by both (x+3) and (x-4), find the values of a and b
Q6: The factors of x3 + x2 – 4x – 4 are (x+2), (x-2) and (x+1). What are the zeros of the polynomials?
Q7: The zeros of the polynomial P(x) = x3 – 6x2 + 11x – 6 are 1, 2 and 3. Write the factors of P(x).
Q8: Show that the polynomial P(x) = 2x3 – 8x2 + 2x + 12 has three distinct real zeros a1= -1, a2= 2 and a3= 3 and express P(x) in the form P(x) = p(x-a1) (x-a2) (x-a3). 13

15. Date: ……………………………….
The Factor Theorem
x3 + 2x2 – 5x – 6 = 0
x3 – 7x + 6 = 0
x3 – x2 – 2x = 0
x3 – 3x2 – 6x + 8 = 0
2x3 + 3x2 – 23x – 12 = 0
x3 + 4x2 – 7x – 10 = 0
Q9: Solve the following cubic equations 14

16. Graphs of polynomial functions
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Graphs of polynomial functions
Q1: what are the zeros of the polynomial : f(X) = (x-1)(x+1)(x-2) ?
using these values and other suitable points sketch the graph of the functions
Q2: consider y= – x3 – x2 +2x which can be written in factored form y= -x (x-1)(x+2).
a) what are the zeros of this polynomial function?
b) by using the zeros of the function and other suitable values of x graph the function.
c) the coefficient of( x3) in this function is negative . What effect does this have on
the shape of the graph ? 15

17. Graphs of polynomial functions
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Graphs of polynomial functions
Q3. Find the zeros of each of the polynomial functions and use them to draw a rough
Sketch graph of the function.
F(x) = x(x-2) (b) F(x) = (x+1)(x-2)2
(c )F(x) = (x-4)(x-1)(3x+2) (d) F(x) = (x+3)2(x-1)
Q4. Factor these quadratic polynomials, find the zeros and sketch their graphs.
(a) f(x) = x2-x (b) F(x) = 2x2-x (c) F(x) = 15+2x-x2 16

18. Graphs of polynomial functions
Date: ……………………………….
Graphs of polynomial functions
Q5: The quartic polynomial function f(x) = x(x+1)(x-1)(x-3) has zeros -1, 0, 1 and 3.
A sketch graph of this function is given which these zeros as the x coordinates of
The points of intersection with the x axis.
Find the zeros of each of the polynomial functions and use them to draw a rough
Sketch graph of the function.
F(x) = (x+2)(x+1)(x-1)(x-3) (c) F(x) = x2(x-2)2
F(x) = x(x+2)(x-1) (d) F(x) = -x(x-5)(x-2)(x+3) 17

19. Relations and Functions
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Relations and Functions 18

20. Relations and Functions
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Relations and Functions 19

21. Relations and Functions
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Relations and Functions 20

22. Date: ……………………………….
Graphing Polynomials 21

23. For the graph of y=f(X) given ,draw sketch of: y=2f(x) Y= ½ f(X)
Date: ……………………………….
Graphing Polynomials
For the graph of y=f(X) given ,draw sketch of:
y=2f(x)
Y= ½ f(X)
Y=f(X)+1
For the graph of y=g(X) given ,draw sketch of:
y=g(x)-2
Y=- g(X)
Y=g(-X) 22

24. 1. Y= - P(x) is its reflection in the x-axis
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Graphing Polynomials
For the curve Y = P(x) :
1. Y= - P(x) is its reflection in the x-axis
2. Y= P(-x) is its reflection in the y-axis
3. The curve Y = P(x) +c can be obtained from the curve Y = P(x) by moving it:
* up c units if c is positive
* down c units if c is negative
4. The curve Y= P(x+c) can be obtained from the curve Y = P(x) by moving it :
* left c units if c is positive
* right c units if c is negative
5. The curve Y = a*P(x) can be obtained from the curve Y= P(x) by :
* stretching P(x) if 0<a<1
* compressing P(x) if a>1
Q1: If Y = P(x). Sketch the graph of P(x) = x3 – 8x2 + 17x – 10
Then by using the graph of P(x) graph each of the following:
i. Y = – P(x) ii. Y = P(-x) iii. Y = P(x) iv. Y = 3 P(x) 23

25. Sketch the graph of P(x) = (x-1)(x-3)(x+1)(x-2)
Date: ……………………………….
Graphing Polynomials
Q2: If Y = P(x)
Sketch the graph of P(x) = (x-1)(x-3)(x+1)(x-2)
Then by using the graph of P(x) graph each of the following:
Y = – P(x)
Y = P(-x)
Y = P(x) -3
Y = 2 P(x) 24

26. Q1: If f(x) = 3x + 2, find the value of:
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Function Notation
Q1: If f(x) = 3x + 2, find the value of:
a. f (0)
b. f (2)
c. f (- 1)
d. f (- 5)
d. f ( ) 1 3
Q2: If g(x) = x – , find the value of:
a. g (1)
b. g (4)
c. g (- 1)
d. g (- 4)
d. g ( ) 4 x 1 2
Q3: If f(x) = 3x – x2 + 2, find the value of:
a. f (0)
b. f (3)
c. f (- 3)
d. f (- 7)
d. f ( ) 3 2 25

27. Q4: If f(x) = 7 – 3x , find in simplest form: a. f (a) b. f (- a)
Date: ……………………………….
Function Notation
Q4: If f(x) = 7 – 3x , find in simplest form:
a. f (a)
b. f (- a)
c. f (a + 3)
d. f (b – 1)
d. f (x + 2)
Q5: If f(x) = 2X2+3X-1 , find in simplest form:
a. f (X+4)
b. f (2-X)
c. f (-X)
d. f (X2)
d. f (x2 -1) 26

28. Find the value of x where G(x) does not exist
Date: ……………………………….
Function Notation
Q6: If
Evaluate
G (2)=
G (0)=
G ( )=
Find the value of x where G(x) does not exist
Find G(x + 2) in simplest form
Find x if G(x) = - 3 1 2 27

29. Q9: If the value of photocopier t years after purchase is given by :
Date: ……………………………….
Function Notation
Q9: If the value of photocopier t years after purchase is given by :
V(t) =9650 – 860t dollars
a) find V(4) and state what V(4) means.
b) find t when V(t)= 5780 and explain what this represents.
c) find the original purchase price of the photocopier. 28

30. Date: ……………………………….
Inverse Functions 29

31. Date: ……………………………….
Inverse Functions 30

32. Date: ……………………………….
Inverse Functions 31

33. Date: ……………………………….
Logarithms 32

34. Date: ……………………………….
Logarithms 33

35. LOGARITHMS IN BASE 10 Logarithms
Date: ……………………………….
Logarithms
LOGARITHMS IN BASE 10
Definition : The logarithm of a positive number, in base 10 , is its power of 10.
For example , since 1000 =103 , we say “ the logarithm of 1000, in base 10 ,is 3” and write this as log =3 or log 1000=3 34

36. Date: ……………………………….
Logarithms 35

37. Date: ……………………………….
Logarithms 36

38. Date: ……………………………….
Laws of Logarithms 37

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Laws of Logarithms 38

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Laws of Logarithms 39

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Laws of Logarithms 40

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Laws of Logarithms 41

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Laws of Logarithms 42

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Laws of Logarithms 43

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Laws of Logarithms 44

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Laws of Logarithms 45

47. Date: ……………………………….
Laws of Logarithms 46

48. Graph y=log10 x Graph y=log2 x Graph y=log5 x Laws of Logarithms 47
Date: ……………………………….
Laws of Logarithms
Graph y=log10 x
Graph y=log2 x
Graph y=log5 x 47

49. Use the graph of y=log x to find the approximate value of: a) Log2=
Date: ……………………………….
Laws of Logarithms
Use the graph of y=log x to find the approximate value of:
a) Log2=
b) log6=
c) Log(0.7)=
d) log(0.2)=
e) log(2.4)=
f) log(1) =
Sketch the graph of: 1) y= log ) y=10x ) y=x on the same axes. 48

50. Sketch the graph of y=log3 and its inverse
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Laws of Logarithms
An entomologist, monitoring a grasshopper plague, notices that the area affected
by the grasshoppers is given by An = 1000 x 2 0.7n hectares, where n is the number
of weeks after the initial observation.
a) use graph to estimate the time taken for the infested area to reach 5000 ha.
b) Find the answer to ( a) using logarithms
The grasshopper problem where the area of infestation
was given by An = 1000 x 20.2n hectares (n is the
number of weeks after initial observation).
a) use graph to estimate the time taken for the infested area to reach 4000 ha.
Sketch the graph of y=log3 and its inverse 49

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