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Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology...

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Solving Cubics Starting Problem solve 0 6 5 2 2 3 x x x Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy to talk about cubics and other New Terminology – P( x ) represents polynomial in x Using the P(x) terminology the problem above can be written as If P(x) = x 3 – 2x 2 – 5x + 6 find the values for x that make such that P(x) = 0
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Page 1: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

solve 0652 23 xxx

Before we start this topic I want to introduce the terminology

P(x) = meaning polynomial in x

This makes it easy to talk about cubics and other polynomials.

New Terminology – P(x) represents polynomial in x

Using the P(x) terminology the problem above can be written as

If P(x) = x3 – 2x2 – 5x + 6 find the values for x that make such that P(x) = 0

Page 2: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Unlike quadratic equations there is no formula available for helping us to solve cubics and as yet we don’t have an algebraic method so we have to go back to

The old Trial and Error method.

Page 3: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Let’s try x = 1

0

6521

615121)1( 23

P

So x = 1 is a solution because it makes the starting equation work out correctly

Page 4: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Let’s try x = 2

So x = 2 is not a solution because it doesn’t satisfy the starting equality

4

61088

625222)2( 23

P

Solutions so far

x = 1

Page 5: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Let’s try x = 3

So x = 3 is a solution because it makes the starting equation work out correctly

0

6151827

635323)3( 23

P

Solutions so far

x = 1

Page 6: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Should we try x = 4

Hint:When using Trial & Error we usually try the whole number factors of the Constant first particularly when the highest order term has a coefficient of 1.

Solutions so far

x = 1 x = 3

Highest order term Constant

Page 7: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Let’s try x = -1

So x = -1 is not a solution because it doesn’t satisfy the starting equality

8

6521

615)1(2)1()1( 23

P

Solutions so far

x = 1 x = 3

Page 8: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

Let’s try x = -2

So x = -2 is a solution because it makes the starting equation work out correctly

0

61088

625)2(2)2()2( 23

P

Solutions so far

x = 1 x = 3

Page 9: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

Starting Problem

0652 23 xxx

So now we have 3 solutions x = 1 x = 3 x = -2Which is all we can expect because the maximum number of solutions for any polynomial = the order (maximum power) of the equation

Page 10: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics

ARRRRGHHHHHH!!!!!!!!!!!!!!!!

There has got to be a quicker method!!

Page 11: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Terminology

Method 1: Use Main Application• Write equation• Highlight it• Go to Interactive then Equation/Inequality then Solve

Method 2: Use Graphs & Tables Application• Enter polynomial as y1 • Draw the graph and make sure all x intercepts are in the

window• Go to Analysis/G solve/Root

Page 12: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Terminology

Method 1: Use Main Application• Write equation• Highlight it• Go to Interactive then Equation/Inequality then Solve

Method 2: Use Graphs & Tables Application• Enter polynomial as y1 • Draw the graph and make sure all x intercepts are in the window• Go to Analysis/G solve/Root

Use two technology methods to find the values for x that make such that P(x) = 0 if:

P(x) = x3 – 2x2 – 5x + 6 x = -2, 1, 3

Page 13: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Algebra

There is also an Algebraic method that can be used if technology is not available and it is sometimes quicker than guessing the answers like we did at the start. It is definitely quicker for situations where your solution involves one or two big numbers and/or fraction answers.

Page 14: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Algebra

The basis of this method involves trying to factorize the polynomial.

0652 23 xxx

0)2)(3)(1( xxx

We are going to look at how we can go from

Which gives us x = 1, x = 3 & x = -2 from our work with quadratics

To

Page 15: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Algebra

The steps to the factorisation (algebraic) method for solving cubics are:

1. Find one factor

2. Use that factor to find its co-factor

3. Factorise the polynomial and deduce the

solutions from the factorised polynomial

Page 16: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Step 1: Find A Factor

Starting Problem

652)(

065223

23

xxxxPlet

xxxSolve

Find A Factor Try x = 1

0

6521

615121)1( 23

P

So x = 1 is a solution (x - 1) is a factor

Page 17: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Step 2: Find Co-Factor

If (x -1) is a factor then

__)__)(__1(

6522

23

xxx

xxx

This is the co-factor and this is what we

have to find

What will be the order of this equation?

Page 18: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Step 2: Find Co-Factor

One method to find a co-factor is to carry out the division

6521

___

23

2

xxxx

xx Which should give

Page 19: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

1

Step 2: Find Co-FactorThe RJ preferred method is to deduce the co-factor

)1(__66

)1(__1

)1(__1

6521

___

2

223

23

2

xx

xxxx

xxxx

xxxx

xx

-1

-6

62 xx

1. Multiply co-factor blank by the factor (they have to multiply out to the polynomial)

2. Deduce co-factor co-efficients using the working space

I call this the co-factor blank

Page 20: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

0)3)(2)(1(

0)6)(1(

06522

23

xxx

xxx

xxx

Step 3: Factorise Polynomial and deduce the solutions

So x = 1 , -2 , 3

Page 21: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Example 2 Step 1: Find A Factor

Example 2

Let’s try x = 1

So x = 1 does not lead to a factor

0863 23 xxx

10

8631

816131)1( 23

P

What are my best options for the next

test

Page 22: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Example 2 Step 1: Find A Factor

Example 2

Let’s try x = -1

0863 23 xxx

So x = -1 is a solution (x + 1) is a factor

0

8631

8)1(6)1(3)1()1( 23

P

Because we already have

the values but not the signs

Page 23: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

1

)1(__88

)1(__22

)1(__1

8631

2

223

23

xx

xxxx

xxxx

xxxx

Example 2: Step 2: Find Co-Factor

Setting out to find the co-factor

2

-8

822 xx

Page 24: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

0)2)(4)(1(

0)82)(1(

08632

23

xxx

xxx

xxx

Example 2: Step 3 - Factorise Polynomial and deduce the solutions

So x = -1 , -4 , 2

Page 25: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Examples To Try

064:3 23 xxxEx

030114:4 23 xxxEx

0252:5 23 xxxEx

02021:6 3 xxEx

02:7 23 xxxEx

08168:8 23 xxxEx

(x + 1)(x – 3)(x – 2) = 0 x = -1, x = 3, x = 2

(x – 2)(x – 5)(x + 3) = 0 x = 2, x = 5, x = -3

(x – 1)(2x + 1)(x – 2) = 0 x = 1, x = -1/2, x = 2

(x – 1)(x + 5)(x – 4) = 0 x = 1, x = -5, x = 4

(x + 2)(x²- x + 1) = 0 x = -2

Page 26: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Examples To Try

0252:5 23 xxxEx

02021:6 3 xxEx

02:7 23 xxxEx

(x – 1)(2x – 1)(x – 2) = 0 x = 1, x = 1/2, x = 2

(x – 1)(x + 5)(x – 4) = 0 x = 1, x = -5, x = 4

(x + 2)(x²- x + 1) = 0 x = -2

53,53,2

0)53)(53)(2(

x

xxx

08168:8 23 xxxEx

Page 27: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Textbook Questions

Exercise 7E p224

Q1 LHS (try RHS if you need extra practice)

Q2 all parts

Page 28: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Some Tricks of The Trade

• For single x terms solve by undoing each process• Do Ex 7E q4 bcde• Read Example 16 • Do Ex 7E q3• Read Example 17• Do Ex 7E q5• When asked to use technology use the graphic

calculator as you did for the quadratics unit.• Do Ex 7E q6

Page 29: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

The Cubic Rules

))((

))((

33)(

33)(

2233

2233

32233

32233

babababa

babababa

babbaaba

babbaaba

Page 30: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

The Cubic Rules

2754368

279233438

33)2(33)2(3)2(

3233)32(

23

23

3223

32233

xxx

xxx

xxx

bxababbaax

6414410827

64163349327

44)3(34)3(3)3(

4333)43(

23

23

3223

32233

xxx

xxx

xxx

bxababbaax

Page 31: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

The Cubic Rules

)964)(32(

)33.2)2)((32(

32))((

3)2(278

2

22

22

333

xxx

xxx

bxabababa

xx

)252016)(54(

)55.4)4)((54(

54))((

5)4(12564

2

22

22

333

xxx

xxx

bxabababa

xx

Page 32: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

The Cubic Rules

)93)(3(2

3))((2

)3(2

)27(2542

2

22

33

33

xxx

bxabababa

x

xx

Page 33: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Questions using Cubic Rules

Ex7D p221

Q4 all parts

Answer

Part A: Using the Cubic Formulae

on the worksheetCubic Equations - Formulae & Technology Exercises

Page 34: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubics With Technology

Answer

Part B: Technology Techniques

on the worksheetCubic Equations - Formulae & Technology Exercises

Page 35: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

The Factor Theorem

)()()(

0)(0

xPoffactoraisbaxsoandxPof

solutionaisa

bthen

a

bPfindweandxPIf

We have been using a version of this for solving cubics

0)()( a

bPthenxPoffactoraisbaxIf

The benefit of this new statement is that it extends our version to cover situations that are not equations such as straight factorisation problems.

Page 36: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Factor Proofs

Example

Without dividing, show that x – 1 is a factor of P(x) = 2x3 – 5x2 + x + 2.

The factor theorem says that P(1) = 0 if x – 1 is a factor of P(x).

P(1) = 2 x 13 – 5 x 12 + 1 + 2 = 2 – 5 + 1 + 2 = 0 so x – 1 is a factor of P(x).

Page 37: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Factor Proofs

Ex 7Dp220

Q1

And for extension try

Q2ab

Q6

Page 38: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Factorising Only Questions

Ex 7D p221

Q3 abde

Page 39: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Divisions With Remainders

1

4332 23

x

xxx

Page 40: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

6

)1(__22

)1(__55

)1(__22

43321

2

223

23

xx

xxxx

xxxx

xxxx

Divisions With Remainders

-5

2

1

6252 2

xxx

2

Page 41: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Remainders

• Do Ex 7B p215

• Q1 LHS

• Q3a

Page 42: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Remainder Theorem

See examples on the boards

Do Ex 7C Q1 LHS Q2

From the Textbook

)()(b

aPisremainderthebaxbydividedisxPWhen

)()(a

bPisremainderthebaxbydividedisxPWhen

Page 43: Solving Cubics Starting Problem solve Before we start this topic I want to introduce the terminology P(x) = meaning polynomial in x This makes it easy.

Solving Cubic Equations Revision


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