Developing Mathematical Habits of Mind in Mathematics Teachers
Unit 1 Mathematical Methods - Teachers' Beehive
Transcript of Unit 1 Mathematical Methods - Teachers' Beehive
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Unit 1 Mathematical Methods
Chapter 6: Polynomials
Objectives
To add, subtract and multiply polynomials.
To divide polynomials.
To use the remainder theorem, factor theorem and rational-root theorem to identify the linear factors of cubic and quartic polynomials.
To solve equations and inequalities involving cubic and quartic polynomials.
To recognise and sketch the graphs of cubic and quartic functions.
To find the rules for given cubic graphs.
To apply cubic functions to solving problems.
To use the bisection method to solve polynomial equations numerically.
6A โ The language of polynomials A Polynomial function follows the rule
๐ = ๐๐๐๐ + ๐๐โ๐๐๐โ๐ + โฆโฆโฆโฆ. ๐๐๐ + ๐๐ ๐ โ ๐ต
Where ๐0, ๐1, โฆ โฆ โฆ โฆ ๐๐ are coefficients.
Degree of a polynomial is the highest power of ๐ฅ with a non-zero coefficient. In summary:
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Example 1
Let ๐(๐ฅ) = 2๐ฅ6 โ ๐ฅ3 + ๐๐ฅ2 + ๐๐ฅ + 20. If ๐(โ1) = 2๐(2) = 0, find the values of ๐ and ๐.
โ The arithmetic of polynomials The operations of addition, subtraction and multiplication for polynomials are naturally defined. The sum, difference and product of two polynomials is a polynomial. Example 2
Let ๐(๐ฅ) = ๐ฅ3 โ 2๐ฅ2 + ๐ฅ, ๐(๐ฅ) = 2 โ 3๐ฅ and โ(๐ฅ) = ๐ฅ2 + ๐ฅ, simplify the following: a) ๐(๐ฅ) + โ(๐ฅ) b) ๐(๐ฅ)โ(๐ฅ)
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โ Equating coefficients Two polynomials ๐ and ๐ are equal only if their corresponding coefficients are equal.
Example 3 The polynomial ๐(๐ฅ) = ๐ฅ3 + 3๐ฅ2 + 2๐ฅ + 1 can be written in the form
(๐ฅ โ 2)(๐ฅ2 + ๐๐ฅ + ๐) + ๐ where ๐, ๐ and ๐ are real numbers. Find the values of ๐, ๐ and ๐.
โ The expansion of (๐ + ๐)๐ We know that (๐ + ๐)2 = ๐2 + 2๐๐ + ๐2 This is called an identity. If we multiply both sides by (๐ + ๐), we obtain
(๐ + ๐)3 = (๐ + ๐)(๐2 + 2๐๐ + ๐2) = ๐3 + 3๐2๐ + 3๐๐2 + ๐3
And so on โฆโฆ.
Complete Exercise 6A Questions page 214
Quadratic function Cubic function Quartic functions
(๐ + ๐)2 = ๐๐ฅ2 + ๐๐ฅ + ๐2, ๐ โ 0 (๐ + ๐)3 = ๐3 + 3๐2๐ + 3๐๐2 + ๐3, ๐ โ 0 (๐ + ๐)4 = ๐๐ฅ4 + 4๐3๐ + 6๐2๐2 + 4๐๐3 + ๐4, ๐ โ 0
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6B โ Division of polynomials
In order to sketch the graphs of many cubic and quartic functions (as well as higher degree polynomials) it is often necessary to find the x-axis intercepts. As with quadratics, finding x-axis intercepts can be done by factorising and then solving the resulting equation using the null factor theorem. All cubic functions will have at least one x-axis intercept. Some will have two and others three.
Examples of long division:
1. Divide ๐ฅ3 โ 4๐ฅ2 โ 11๐ฅ + 30 by ๐ฅ โ 2.
2. Divide 2๐ฅ3 โ 6๐ฅ2 โ 10๐ฅ + 25 by ๐ฅ2 โ 2
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Complete Exercise 6B Questions page 219
6C โ Factorisation of polynomials
โ Remainder theorem and factor theorem
Examples :
1. Find the remainder when ๐(๐ฅ) = 3๐ฅ4 โ 9๐ฅ2 + 27๐ฅ โ 8 is divided by ๐ฅ โ 2.
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2. Factorise ๐(๐ฅ) = ๐ฅ3 โ 4๐ฅ2 โ 11๐ฅ + 30 by ๐ฅ โ 2 and hence solve for ๐ฅ.
3. Given ๐ฅ + 1 and ๐ฅ โ 2 are factors of 6๐ฅ4 โ ๐ฅ3 + ๐๐ฅ2 โ 6๐ฅ + ๐, find ๐ and ๐.
โ Sums and differences
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Examples:
1. Factorise 27๐ฅ3 โ 1 2. Factorise 8๐3 + 125๐3
โ The rational-root theorem
Examples:
Use the rational-root theorem to factorise ๐(๐ฅ) = 3๐ฅ3 + 8๐ฅ2 + 2๐ฅ โ 5.
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โ Alternative method for division of polynomials
Synthetic division is an alternative method to long division that can be used when factorising a polynomial, and the method is not the priority.
Complete Exercise 6C Questions page 227
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6D โ Solving cubic equations
In order to solve a cubic equation, the first step is often to factorise. Apply the same rule(s) practiced in Exercise 6C to find a factor(s) or by other means, then equate to zero to find solutions. Examples:
1. Solve each of the following: a) 2(๐ฅ โ 1)3 = 32
b) 2๐ฅ3 โ ๐๐ฅ2 โ 22๐ฅ + 11๐
2. Solve 2๐ฅ3 โ 5๐ฅ2 + 5๐ฅ โ 2 = 0
Complete Exercise 6D Questions page 231
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6E โ Cubic functions of the form ๐(๐) = ๐(๐ โ ๐)๐ + ๐ In Chapter 3 we saw that all quadratic functions can be written in โturning point formโ and that the graphs of all quadratics have one basic form, the parabola. This is not true of cubic functions. Consider cubic functions in the form ๐(๐ฅ) = ๐(๐ฅ โ โ)3 + ๐. The graphs of these functions can be formed by simple transformations of the graph of ๐(๐ฅ) = ๐ฅ3. For example, the graph of ๐(๐ฅ) = (๐ฅ โ 1)3 + 3 is obtained from the graph of ๐(๐ฅ) = ๐ฅ3 by a translation of 1 unit in the positive direction of the ๐ฅ-axis and 3 units in the positive direction of the ๐ฆ-axis.
โ Transformations of the graph ๐(๐) = ๐๐ Dilations from an axis and reflections in an axis to cubic functions in the form ๐(๐) = ๐๐๐
It should be noted that the implied domain of all cubics is โ and the range is also โ. The point of inflection can move as described in the above function ๐(๐ฅ) = (๐ฅ โ 1)3 + 3
General form:
Sketch the graph
This point is called the point of inflection (a point of
zero gradient)
๐(๐ฅ)
๐ฅ
Point of inflection is (1,3)
Dilation factor Horizontal translation
Vertical translation
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โ The function ๐: โ โ โ, ๐(๐) = ๐๐
๐ The functions with rules of the form ๐(๐ฅ) = ๐(๐ฅ โ โ)3 + ๐ are one-to-one functions. Hence each of these functions has an inverse function.
The inverse function of ๐(๐ฅ) = ๐ฅ3 is ๐โ1(๐ฅ) = ๐ฅ1
3. Examples:
1. Sketch the graph of ๐(๐ฅ) = โ2(๐ฅ + 1)3 + 1. Show all intercepts andpoint of inflection.
2. Find the inverse functin of ๐(๐ฅ) = โ2(๐ฅ + 1)3 + 1.
๐(๐ฅ)
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Complete Exercise 6E Questions page 235
6F โ Graphs of factorised cubic functions The general cubic function written in polynomial form is
๐ = ๐๐๐ + ๐๐๐ + ๐๐ + ๐ The graph of a cubic function can have one, two or three ๐ฅ-axis intercepts.
โ If a cubic can be written as the product of three linear factors, ๐ฆ = ๐(๐ฅ โ ๐ผ)(๐ฅ โ ๐ฝ)(๐ฅ โ ๐พ)
then its graph can be sketched by following these steps: โช Find the ๐ฆ-axis intercept. โช Find the ๐ฅ-axis intercepts. โช Prepare a sign diagram. โช Consider the ๐ฆ-values as ๐ฅ increases to the right of all ๐ฅ-axis intercepts. โช Consider the ๐ฆ-values as ๐ฅ decreases to the left of all ๐ฅ-axis intercepts.
โ If there is a repeated factor to the power 2, the ๐ฆ-values have the same sign
immediately to the left and right of the corresponding ๐ฅ-axis intercept. Linear factors will be written as ๐ฆ = ๐(๐ฅ โ ๐ผ)(๐ฅ โ ๐ฝ)2
Graphs of some cubic functions
Points of
inflection
Repeated factors
Three intercepts, no repeated factors
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โ Sign diagrams A sign diagram is a number-line diagram which shows when an expression is positive or negative. The following is a sign diagram for a cubic function, the graph of which is also shown.
Using a sign diagram requires that the factors, and the ๐ฅ-axis intercepts, be found. The ๐ฆ โaxis intercept and sign diagram can then be used to complete the graph. Example:
1. Sketch the graph of ๐ฆ = (๐ฅ + 1)(๐ฅ โ 2)(๐ฅ + 3). Hint: draw a sign diagram first.
2. Sketch the graph of ๐ฆ = 3๐ฅ3 โ 4๐ฅ2 โ 13๐ฅ โ 6
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6G โ Solving cubic inequalities As was done for quadratic inequalities, we can solve cubic inequalities by considering the graph of the corresponding polynomial. Examples:
Solve the following cubic inequalities. a) (๐ฅ + 1)(๐ฅ โ 2)(๐ฅ + 3) โฅ 0
b) (๐ฅ + 1)(๐ฅ โ 2)(๐ฅ + 3) < 0
Complete Exercise 6G Questions page 242
6H โ Families of cubic polynomial functions โ Finding rules for cubic polynomialfunctions.
Apply the same rules that were applied to finding the rule for a quadratic polynomial function. Use any of the general rules: ๐(๐ฅ) = ๐(๐ฅ โ โ)3 + ๐ Or ๐(๐ฅ) = ๐๐ฅ3 + ๐๐ฅ2 + ๐๐ฅ + ๐ Example:
1. Find a cubic function whose graph touches the ๐ฅ-axis at ๐ฅ = โ4, cuts it at the origin, and has a value 6 when ๐ฅ = โ3.
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2. Find the equation of the following cubic in the form ๐ฆ = ๐๐ฅ3 + ๐๐ฅ2.
Complete Exercise 6H Questions page 246
6I โ Quartic and other polynomial functions
โ Quartic functions of the form ๐(๐) = ๐(๐ โ ๐)๐ + ๐ โช As with other graphs it has been seen that changing the value of ๐ simply narrows or
broadens the graph without changing its fundamental shape.
โช If ๐ < 0, the graph is inverted. โช The significant feature of the graph of a quartic of this
form is the turning point. โช The turning point of ๐ฆ = ๐ฅ4 is at the origin (0,0). โช For the graph of a quartic function of the form
๐(๐) = ๐(๐ โ ๐)๐ + ๐, the turning point is at (๐, ๐). When sketching quartic graphs of the form ๐ฆ = ๐(๐ฅ โ โ)4 + ๐, first identify the turning point. To add further detail to the graph, find the ๐ฅ-axis and ๐ฆ-axis intercepts. Examples of possible cubic functions and the shape of their graphs.
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โ Odd and even polynomials Knowing that a function is even or that it is odd is very helpful when sketching its graph.
โ A function ๐ is even if ๐(โ๐ฅ) = ๐(๐ฅ). This means that the graph is symmetric about the ๐ฆ-axis. That is, the graph appears the same after reflection in the ๐ฆ-axis.
โ A function ๐ is odd if ๐(โ๐ฅ) = โ๐(๐ฅ). The graph of an odd function has rotational symmetry with respect to the origin: the graph remains unchanged after rotation of 180ยฐ about the origin. A power function is a function ๐ with rule ๐(๐ฅ) = ๐ฅ๐ where ๐ is a non-zero real number. We will only consider the cases where ๐ is a positive integer or ๐ โ {โ2, โ1,12,13}.
Even-degree power functions
The functions with rules ๐(๐ฅ) = ๐ฅ2 and ๐(๐ฅ) = ๐ฅ4 are examples of even-degree power functions. The following are properties of all even-degree power functions: ๐(โ๐ฅ) = (๐ฅ) for all ๐ฅ ๐(0) = 0
As ๐ฅ โ ยฑโ, ๐ฆ โ โ. Odd-degree power functions The functions with rules ๐(๐ฅ) = ๐ฅ3 and ๐(๐ฅ) = ๐ฅ5are examples of odd-degree power functions. The following are properties of all odd-degree power functions: ๐(โ๐ฅ) = โ๐(๐ฅ) for all xx ๐(0) = 0
As ๐ฅ โ โ, ๐ฆ โ โ and
as ๐ฅ โ โโ, ๐ฆ โ โโ. Note: if a function has both even and odd powers, it is neither even or odd. Example:
3. State whether the following polynomials is even or odd. a) ๐(๐ฅ) = 3๐ฅ4 โ ๐ฅ2
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b) ๐(๐ฅ) = 8๐ฅ3 + 5๐ฅ โ 9
4. a) On the same set of axis sketch the graphs of ๐(๐ฅ) = ๐ฅ4 and ๐(๐ฅ) = 9๐ฅ2. b) Solve the equation ๐(๐ฅ) = ๐(๐ฅ) c) Solve the inequality ๐(๐ฅ) โค ๐(๐ฅ)
Complete Exercise 6I Questions page 252
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6J โ Applications of polynomial functions Textbook examples:
1. A square sheet of tin measures 12 ๐๐ ร 12 ๐๐.
Four equal squares of edge ๐ฅ cm are cut out of the corners and the sides are turned up to form an open rectangular box. Find:
a) the values of ๐ฅ for which the volume is 100 ๐๐3 b) the maximum volume.
Solution The figure shows how it is possible to form many open rectangular boxes with dimensions
12 โ 2๐ฅ, 12 โ 2๐ฅ and ๐ฅ.
The volume of the box is
๐ = ๐ฅ(12 โ 2๐ฅ)2, 0 โค ๐ฅ โค 6 which is a cubic model. We complete the solution using a CAS calculator as follows.
2. It is found that 250 metres of the path of a stream can be modelled by a cubic function. The cubic passes through the points (0,0), (100,22), (150,โ10), (200,โ20).
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a) Find the equation of the cubic function. b) Find the maximum deviation of the graph from the ๐ฅ-axis for ๐ฅ โ [0,250].
Complete Exercise 6J Questions page 255
Chapter 6 Checklist This is the minimum requirement, do more where you feel it is necessary.
Exercise 6A: The language of polynomials Q 2, 3, 4, 5a,c,d,h, 6, 7, 8, 9, 10b
Exercise 6B: Division of polynomials Q 1a,c, 2b,d, 3b,c, 4a,d, 5, 6, 7a,c,e,f
Exercise 6C: Factorisation of polynomials Q 1-LHS, 2,b,c, 3, 4, 5-RHS, 7a,d, 8b,c,e,g,h, 9,a,c, 10
Exercise 6D: Solving cubic equations Q 1b,d, 2a,c, 3c,d, 4-RHS, 5-b,c,f, 6a,c,e, 7b,d
Exercise 6E: Cubic functions f(x)=a(x-h)3+k Q 2, 3, 4
Exercise 6F: Graphs of factorised cubic functions Q1a,c, 2b,e,f, 3a,c,d, 4, 5
Exercise 6G: Solving cubic inequalities Q 1, 2
Exercise 6H: Families of cubic polynomials Q 1, 3, 5, 6, 7a,c, 8 9a,c,g
Exercise 6I: Quartic and other polynomials Q 1a,d, 2a,d,e,f, 3-RHS, 4-LHS, 5, 6, 8
Exercise 6J: Applications of polynomials Q- All
Chapter 6 Review Questions All