Remaindor Theorem

1.     If F(x) = 2x + 8/x², calculate F(2) and F(-2)

2.     If F(x) = -12/x², calculate F(2)

3.     If P(x) = 2x³ + 3x² -3x -2 evaluate P(1) and P(-2)

4.     If P(x) = x³ –x² + 2x + 4 evaluate P(-1) and P(2)

5.     If P(x) = ax³ + bx² -3x -2 and P(1) = 0 and P(-2) = 0, find the values of a and b.

6.     If P(x) = x³ –x² + ax + b and P(-1) = 0 and P(2) = 12, find the values of a and b.

Proving Trig Identity

2 things to remember

1. sin q and cosq are base trigs
2. sin² q + cos² q = 1

Must know trigs in terms of base

tan q = sinq

             cosq

cot q = cosq          

             sinq

sec q = __1_

             cosq

csc q = __1__

             sinq

 

Prove the identity

 

       1         -  sin q  = cot q  cos q                             

   sin  q                                                                                     

Prove the identity

                            sin 2q           =   tan q

1 + cos2q 

 

Prove the following

1.    sec²x + cosec²x = sec²x cosec²x

2.    sec x cot x = cosec x

3.    cos²x__    = 1 + sin x

1 – sin x

4.    sin x + sin x cos²x = cosec x

5.    sin x cos x tan x = 1 – cos²x

6.    tan x + cot x = sec x cosec x

7.    1 + cos x        = sin x___

sin x                    1 – cos x

                   

Test 1 Differentiation

1.   The function y = x³ + ax² – 7x -1 has a stationary value where x = 1. Find the value of a. Find the other stationary points and the nature of these points?

Question 1          When you see find the value of a, what must you expect

Question 2          What does stationary means?

Question 3          What is the bread for this stationary value

Question 4          Differentiate y

Question 5          What is the linear equation

Question 6                    What is the maximum number of possible stationary points

Question 7          How would you find the other stationary point(s)

Question 8          What is meant by the nature of the points

Question 9          How you do find the nature of the stationary points

Question 10       Find the nature of all stationary points

Calculate the gradient of the curve y = x - 4

                                                                      x - 1

 at the point where it crosses the x- axis

Question 1

What happens when a point crosses the x-axis

Question 2

When you see the word Gradient, what should you look for

Question 3

Differentiate y

Question 4

Find the required gradient

Quiz 1 Remaindor Theorem

1.   F(x) = x³  + ax² + bx + 6

The remainder when F(x) is divided by x + 1 and x – 2 are 20 and 8 respectively, find the values of a and b. Hence solve the equation F(x) = 0

Question 1              

When you see find the values of a and b, what does that tell you that you must get in order to solve?

Question 2

What does the remainder when F(x) is divided by x + 1 is 20 mean?

Question 3   What does the remainder when F(x) is divided by x – 2 is 8 mean?

Question 4               Get 1^st equation

Question 5               Get 2^nd equation

Question 6               When solve is given, what 2 ways do you expect

Question 7               How to solve this equation

Show that 2x³ + x² – 13x + 6 is divisible by x – 2 and hence find the other factors of the expression

Question 1

What does divisible by x – 2 means

Question 2

How would you use Qu 1 information and obtain the result

Question 3

How would you find the other factors?

Question 4

Find all the factors

Solving a cubic

To solve a cubic, a factor must be given or determined.
Then the equation is divided by polynomial division by the factor.
The answer will be a quadratic and the remaindor 0.
Solve the quadratic for the other 2 answers.

Remember a factor or a root or divisible means remaindor is 0.
Use Remaindor Theorem to verify the remaindor is 0.
If first factor has to be determined, try (x - 1) or (x + 1) or (x - 2) or (x +2) for remaindor of 0.

p(x) = x³ – 7x – 6, and let's divide by the linear factor x – 4 (so a = 4): So we get a quotient of q(x) = x² + 4x + 9 on top, with a remainder of r(x) = 30

1. Show that 2x³ + x² -13x + 6 is divisible by x-2. Hence solve the equation.
2. Find the value of the constant k such that (x + 1) is a factor of the expression  2x³ + 7x² + kx - 3. For this value of k solve the equation  2x³ + 7x² + kx - 3.
3. f(x) = x³ + ax² + bx + 6 The remainder when f(x) is divided by x + 1 and x - 2 are 20 and 8 respectively, find the values of a and b. Hence solve the equation f(x) = 0

The word solve

At this level solve can only apply for

• linear ie 1 unknown and the highest power of unknown is 1
• quadratic ie 1 unknown and the highest power of unknown is 2
• cubic ie 1 unknown and the highest power of unknown is 3

Solving linear is one side of = is unknown and other is the known
Solving quadratic

1. Must be = 0
2. Replace the appliance with a ( )
3. Use quadratic formula using the calculator

1.   Solve the equation 2x² + 13x + 6 = 0

2.   Solve the equation 3x² – 10x = 25 

3.   Solve the equation x² – 7x + 12 = 0

4.   Solve the equation x² + 13x + 42 = 0

5.   Solve the equation 6x² + x  - 35 = 0

6. Solve the equation 2cos²x – 3 cos x –2 = 0

1.      Solve 3x² – 10x + 3 = 0

2.      Solve 3(2ⁿ)² – 10(2ⁿ) + 3 = 0

3.      Solve 3(2²ⁿ) – 10(2ⁿ) + 3 = 0

4.      Solve 3(4ⁿ) – 10(2ⁿ) + 3 = 0

5.      Solve 3(tan x)² – 10(tan x) + 3 = 0

Seeing the connectors

How many connectors and terms in the following and group like terms:

1.      A = 8r -2r² – ½ πr²

2.      f(x) = 2x³ -9x² + Ax + B, A and B are constants.

3.      y = 2x³ – x² + 4x

4.      log(3x + 2) + 6 log2 = 2 + log (2x + 1)

5.      log (x – 8) + log (9/2) = 1 + log(x/4)

6.    2tan²z + 11sec z + 7 = 0

7.    4sin2 x = 6 – 9cos x

8.    3cos y + cot y = 0

9.    cos 2x = 2 cos x

10.cot x sin 2x = 1 + cos 2x

11.sin 3x = 3 sin x - 4 sin³ x

1.   2x² + 13x + 6 = 0

2.   3x² – 10x – 25 = 0

3.   x² – 7x + 12 = 0

4.   x² + 13x + 42 = 0

5.   6x² + x  - 35 = 0

P(x) = 2x³ +  3x² - ax + b
P(x) = x³ - x² +  ax +  b