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11 x1 t11 04 chords of a parabola (2012)
The document discusses chords of a parabola. It defines a chord as a line segment connecting two points on a parabola. It shows that the slope of any chord can be expressed as a simple formula involving the x-intercepts of the chord points. It also proves that the slope of a focal chord, which passes through the focus, is always 1/2.
X2 t08 02 induction (2012)
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
11 x1 t13 04 converse theorems (2012)
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
11 x1 t11 01 locus (2012)
The document defines locus as the collection of all points whose location is determined by some stated law. It provides examples of finding the locus of points that are:
1) Always 4 units from the origin, which is a circle with radius 4 centered at the origin.
2) Always 5 units from the y-axis, which is the line y=±5.
3) Always 3 units from the line y=x+1, which is the pair of parallel lines that are a distance of 3 units above and below the given line.
11 x1 t02 06 relations & functions (2013)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
11 x1 t10 02 critical points (2012)
Critical points occur when the derivative of a function is equal to zero or undefined. This includes local maxima and minima where the derivative is zero, points of inflection where the second derivative is zero, and places where the curve is vertical or has a "cusp" where the derivative is undefined.
Recommended
11 x1 t11 04 chords of a parabola (2012)
The document discusses chords of a parabola. It defines a chord as a line segment connecting two points on a parabola. It shows that the slope of any chord can be expressed as a simple formula involving the x-intercepts of the chord points. It also proves that the slope of a focal chord, which passes through the focus, is always 1/2.
X2 t08 02 induction (2012)
The document discusses mathematical induction. It first proves that the sum of the series from 1/2 to 1/n is less than or equal to 2 - 1/n. It then proves by induction that for a sequence defined by a1 = 2 and an+1 = 2 + an, an is always less than 2 for n ≥ 1.
11 x1 t13 04 converse theorems (2012)
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
11 x1 t11 01 locus (2012)
The document defines locus as the collection of all points whose location is determined by some stated law. It provides examples of finding the locus of points that are:
1) Always 4 units from the origin, which is a circle with radius 4 centered at the origin.
2) Always 5 units from the y-axis, which is the line y=±5.
3) Always 3 units from the line y=x+1, which is the pair of parallel lines that are a distance of 3 units above and below the given line.
11 x1 t02 06 relations & functions (2013)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
11 x1 t10 02 critical points (2012)
Critical points occur when the derivative of a function is equal to zero or undefined. This includes local maxima and minima where the derivative is zero, points of inflection where the second derivative is zero, and places where the curve is vertical or has a "cusp" where the derivative is undefined.
11 X1 T01 03 factorising (2010)
The document provides steps for factorising expressions:
1) Look for common factors and divide them out
2) Factorise the difference of two squares using the form (a-b)(a+b)
3) Factorise quadratic trinomials into the product of two binomials using the forms x2 + (a+b)x + ab or (x+a)(x+b)
Examples are provided for each type of factorisation.
11X1 T10 09 mathematical induction 2
The document discusses mathematical induction and provides examples of using it to prove statements. It introduces the concept of mathematical induction, which involves three steps: 1) proving the statement is true for n=1, 2) assuming it is true for an integer k, and 3) proving it is true for k+1. The document then works through two examples - proving that n(n+1)(n+2) is divisible by 3, and proving that 3^n + 2^(n+2) is divisible by 5.
11 X1 T05 01 Division Of An Interval
The document discusses coordinate geometry concepts including the distance formula and midpoint formula. It explains that the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint. It also discusses dividing intervals, noting that for a level 2 math exam it is restricted to midpoint divisions in a 1:1 ratio, while an extension 1 exam can involve any ratio for internal or external divisions. Examples are provided to illustrate the concepts.
12 X1 T04 04 Growth and Decay (2010)
The document discusses exponential growth and decay models, where the growth or decay rate of a population is proportional to the population. It provides examples of using exponential functions to model bacterial population growth rates over time and calculating the annual growth rate and half-life of a human population on an island based on census data from 1960 and 1970.
11X1 T14 07 approximations
The document describes the trapezoidal rule for approximating the area under a curve. The trapezoidal rule works by dividing the area into trapezoid sections and summing their individual areas. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the internal y-values, divided by the number of sections. An example applies this to estimate the area under a given curve divided into 4 intervals.
11 x1 t04 05 sine rule (2013)
The document discusses the sine rule and formulas for solving triangles. It introduces the sine rule, which states that in any triangle ABC, a/sinA = b/sinB = c/sinC. It then shows examples of using the sine rule to calculate unknown side lengths and heights of triangles. The document also covers the formula for calculating the area of a triangle as 1/2 * base * height or 1/2 * ab * sinC.
11 x1 t04 01 trigonometric ratios (2012)
The document defines trigonometric ratios such as sine, cosine, and tangent using a right triangle. It provides examples of calculating trigonometric ratios given angle measures. It also gives exact trigonometric ratios for 30°, 45°, and 60° angles.
11 x1 t12 03 second derivative (2013)
The document discusses the second derivative and provides examples of taking the second derivative of various functions. Some key points covered include:
- The notation for the second derivative is f''(x)
- An example is provided of taking the second derivative of the function f(x) = x^2 + x + 1
- Additional examples show taking the second and third derivatives of other functions
12 x1 t03 01 arcs & sectors (2013)
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
X2 T02 01 multiple roots
The document discusses properties of polynomials. It states that if a factor (x - a) divides a polynomial P(x), then P(a) = 0. It also proves that if a polynomial P(x) has a root x = a with multiplicity m, then the derivative of P(x), P'(x), will have a root x = a with multiplicity m - 1. Examples are provided to illustrate these concepts.
11X1 T15 02 finance formulas V2
The document contains formulae for simple interest, compound interest, depreciation, and investing money through regular instalments. It provides the formulae, explanations of the variables, and examples of how to apply the formulae to calculate interest earned, future value, depreciated value, and retirement savings from regular contributions.
11 x1 t02 03 surds (2012)
The document discusses surds, which are irrational numbers containing a radical symbol that cannot be calculated exactly. It defines surds and provides laws for manipulating them, such as multiplying or taking powers of surds. Examples are given to illustrate how to apply the laws and perform operations like addition and multiplication with surds. Conjugate surds are also introduced as a concept.
11X1 T12 09 locus problems (2011)
This document provides instructions for solving locus problems by eliminating parameters (p's and q's) from coordinate expressions. There are three types of problems: 1) No parameters in x or y, 2) Obvious relationship between x and y involving one parameter, 3) Non-obvious relationship requiring a previously proven relationship between parameters. Examples of each type are provided, along with sample locus problems involving points on parabolas and their tangents/normals.
11X1 T11 03 parametric coordinates (2010)
The document discusses parametric coordinates and compares them to Cartesian coordinates. It states that parametric coordinates describe a curve using two equations where points are defined by a single parameter, while Cartesian coordinates use a single equation where points are defined by two numbers. The document then provides an example of a parabola with the parametric coordinates x=2at, y=at^2 and discusses how to derive the Cartesian equation from the parametric form. It also gives the focus of the parabola as (1/4,0) and calculates the parametric coordinates of the curve y=8x^2 as (t/16, t^2/32).
11 x1 t16 07 approximations (2012)
The document discusses approximations of areas under curves using the trapezoidal rule. It shows the general trapezoidal rule formula for approximating the area under a curve between two points by dividing the area into trapezoids. An example is provided to demonstrate applying the trapezoidal rule with 4 intervals to estimate the area under the curve y = (4 - x)^(1/2) from x = 0 to x = 2, giving an answer of 2.996 units^2 correct to 3 decimal places.
11 x1 t15 05 polynomial results (2013)
The document discusses properties of polynomials:
1) If a polynomial P(x) has k distinct real zeros a1, a2,..., ak, then (x - a1)(x - a2)...(x - ak) is a factor of P(x).
2) If a polynomial has degree n and n distinct real zeros, then it can be written as (x - a1)(x - a2)...(x - an).
3) A polynomial of degree n cannot have more than n distinct real zeros.
An example shows a polynomial with a double zero at -7 and single zero at 2 can be written as (x - 2)(x + 7)2.
11 x1 t14 05 sum of an arithmetic series (2012)
The document discusses the formula for calculating the sum of an arithmetic series. It provides the general formula as Sn = (a + l)n/2 if the last term (l) is known, and as Sn = (2a + (n-1)d)/2 otherwise. It also gives examples of using the formula to calculate specific sums.
11X1 T14 04 areas
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
X2 t02 01 multiple roots (2102)
The document discusses polynomials and their properties. It contains three key points:
1) If a factor of a polynomial P(x) is (x - a), then P(a) = 0. This means that if x = a is a root of the polynomial, then the polynomial is equal to 0 at that value.
2) If a polynomial P(x) has a root x = a with multiplicity m, then the derivative of P(x), P'(x), will have a root x = a with multiplicity m - 1. This is proven using properties of derivatives.
3) An example problem is worked through to demonstrate finding the double root of a polynomial using the previous property
Goodbye slideshare UPDATE
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
Goodbye slideshare
NOTE: there is no sound, i.e. it is not a slidecast, as slideshare are going to get rid of this feature.
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11 X1 T01 03 factorising (2010)
The document provides steps for factorising expressions:
1) Look for common factors and divide them out
2) Factorise the difference of two squares using the form (a-b)(a+b)
3) Factorise quadratic trinomials into the product of two binomials using the forms x2 + (a+b)x + ab or (x+a)(x+b)
Examples are provided for each type of factorisation.
11X1 T10 09 mathematical induction 2
The document discusses mathematical induction and provides examples of using it to prove statements. It introduces the concept of mathematical induction, which involves three steps: 1) proving the statement is true for n=1, 2) assuming it is true for an integer k, and 3) proving it is true for k+1. The document then works through two examples - proving that n(n+1)(n+2) is divisible by 3, and proving that 3^n + 2^(n+2) is divisible by 5.
11 X1 T05 01 Division Of An Interval
The document discusses coordinate geometry concepts including the distance formula and midpoint formula. It explains that the distance formula calculates the length of the hypotenuse between two points using Pythagoras' theorem. The midpoint formula averages the x- and y-coordinates of two points to find the midpoint. It also discusses dividing intervals, noting that for a level 2 math exam it is restricted to midpoint divisions in a 1:1 ratio, while an extension 1 exam can involve any ratio for internal or external divisions. Examples are provided to illustrate the concepts.
12 X1 T04 04 Growth and Decay (2010)
The document discusses exponential growth and decay models, where the growth or decay rate of a population is proportional to the population. It provides examples of using exponential functions to model bacterial population growth rates over time and calculating the annual growth rate and half-life of a human population on an island based on census data from 1960 and 1970.
11X1 T14 07 approximations
The document describes the trapezoidal rule for approximating the area under a curve. The trapezoidal rule works by dividing the area into trapezoid sections and summing their individual areas. In general, the area is approximated as the average of the initial and final y-values plus twice the sum of the internal y-values, divided by the number of sections. An example applies this to estimate the area under a given curve divided into 4 intervals.
11 x1 t04 05 sine rule (2013)
The document discusses the sine rule and formulas for solving triangles. It introduces the sine rule, which states that in any triangle ABC, a/sinA = b/sinB = c/sinC. It then shows examples of using the sine rule to calculate unknown side lengths and heights of triangles. The document also covers the formula for calculating the area of a triangle as 1/2 * base * height or 1/2 * ab * sinC.
11 x1 t04 01 trigonometric ratios (2012)
The document defines trigonometric ratios such as sine, cosine, and tangent using a right triangle. It provides examples of calculating trigonometric ratios given angle measures. It also gives exact trigonometric ratios for 30°, 45°, and 60° angles.
11 x1 t12 03 second derivative (2013)
The document discusses the second derivative and provides examples of taking the second derivative of various functions. Some key points covered include:
- The notation for the second derivative is f''(x)
- An example is provided of taking the second derivative of the function f(x) = x^2 + x + 1
- Additional examples show taking the second and third derivatives of other functions
12 x1 t03 01 arcs & sectors (2013)
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
X2 T02 01 multiple roots
The document discusses properties of polynomials. It states that if a factor (x - a) divides a polynomial P(x), then P(a) = 0. It also proves that if a polynomial P(x) has a root x = a with multiplicity m, then the derivative of P(x), P'(x), will have a root x = a with multiplicity m - 1. Examples are provided to illustrate these concepts.
11X1 T15 02 finance formulas V2
The document contains formulae for simple interest, compound interest, depreciation, and investing money through regular instalments. It provides the formulae, explanations of the variables, and examples of how to apply the formulae to calculate interest earned, future value, depreciated value, and retirement savings from regular contributions.
11 x1 t02 03 surds (2012)
The document discusses surds, which are irrational numbers containing a radical symbol that cannot be calculated exactly. It defines surds and provides laws for manipulating them, such as multiplying or taking powers of surds. Examples are given to illustrate how to apply the laws and perform operations like addition and multiplication with surds. Conjugate surds are also introduced as a concept.
11X1 T12 09 locus problems (2011)
This document provides instructions for solving locus problems by eliminating parameters (p's and q's) from coordinate expressions. There are three types of problems: 1) No parameters in x or y, 2) Obvious relationship between x and y involving one parameter, 3) Non-obvious relationship requiring a previously proven relationship between parameters. Examples of each type are provided, along with sample locus problems involving points on parabolas and their tangents/normals.
11X1 T11 03 parametric coordinates (2010)
The document discusses parametric coordinates and compares them to Cartesian coordinates. It states that parametric coordinates describe a curve using two equations where points are defined by a single parameter, while Cartesian coordinates use a single equation where points are defined by two numbers. The document then provides an example of a parabola with the parametric coordinates x=2at, y=at^2 and discusses how to derive the Cartesian equation from the parametric form. It also gives the focus of the parabola as (1/4,0) and calculates the parametric coordinates of the curve y=8x^2 as (t/16, t^2/32).
11 x1 t16 07 approximations (2012)
The document discusses approximations of areas under curves using the trapezoidal rule. It shows the general trapezoidal rule formula for approximating the area under a curve between two points by dividing the area into trapezoids. An example is provided to demonstrate applying the trapezoidal rule with 4 intervals to estimate the area under the curve y = (4 - x)^(1/2) from x = 0 to x = 2, giving an answer of 2.996 units^2 correct to 3 decimal places.
11 x1 t15 05 polynomial results (2013)
The document discusses properties of polynomials:
1) If a polynomial P(x) has k distinct real zeros a1, a2,..., ak, then (x - a1)(x - a2)...(x - ak) is a factor of P(x).
2) If a polynomial has degree n and n distinct real zeros, then it can be written as (x - a1)(x - a2)...(x - an).
3) A polynomial of degree n cannot have more than n distinct real zeros.
An example shows a polynomial with a double zero at -7 and single zero at 2 can be written as (x - 2)(x + 7)2.
11 x1 t14 05 sum of an arithmetic series (2012)
The document discusses the formula for calculating the sum of an arithmetic series. It provides the general formula as Sn = (a + l)n/2 if the last term (l) is known, and as Sn = (2a + (n-1)d)/2 otherwise. It also gives examples of using the formula to calculate specific sums.
11X1 T14 04 areas
The document discusses calculating the area below the x-axis (A) for different functions f(x). It shows that A is given by the integral of f(x) from the left bound to the right bound, or equivalently the negative integral from the right bound to the left bound. As an example, it calculates A for the function f(x)=x^3 from -1 to 1, showing A = 1/2. It also notes that for odd functions, A can be calculated as half the integral from 0 to 1.
X2 t02 01 multiple roots (2102)
The document discusses polynomials and their properties. It contains three key points:
1) If a factor of a polynomial P(x) is (x - a), then P(a) = 0. This means that if x = a is a root of the polynomial, then the polynomial is equal to 0 at that value.
2) If a polynomial P(x) has a root x = a with multiplicity m, then the derivative of P(x), P'(x), will have a root x = a with multiplicity m - 1. This is proven using properties of derivatives.
3) An example problem is worked through to demonstrate finding the double root of a polynomial using the previous property
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More from Nigel Simmons
Goodbye slideshare UPDATE
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
Goodbye slideshare
NOTE: there is no sound, i.e. it is not a slidecast, as slideshare are going to get rid of this feature.
11 x1 t01 03 factorising (2014)
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
11 x1 t01 01 algebra & indices (2014)
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
12 x1 t01 01 log laws (2013)
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
X2 t02 03 roots & coefficients (2013)
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
X2 t02 02 multiple roots (2013)
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
X2 t02 01 factorising complex expressions (2013)
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
11 x1 t16 07 approximations (2013)
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
11 x1 t16 05 volumes (2013)
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
11 x1 t16 04 areas (2013)
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
11 x1 t16 02 definite integral (2013)
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
More from Nigel Simmons (20)
12 x1 t01 03 integrating derivative on function (2013)
12 x1 t01 03 integrating derivative on function (2013)
11 x1 t14 07 sum & diference of powers (2102)
- 1. Sum & Difference Of Powers
- 2. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
- 3. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1
- 4. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 NOTE: n must be odd
- 5. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 NOTE: n must be odd e.g. x 5 32
- 6. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 NOTE: n must be odd e.g. x 5 32 x 5 25
- 7. Sum & Difference Of Powers a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 a n b n a b a n1 a n2b a n3b 2 a 2b n3 ab n2 b n1 NOTE: n must be odd e.g. x 5 32 x 5 25 x 2x 4 2 x 3 4 x 2 8 x 16
- 8. Exercise 6M; 1dhlo, 2bdf, 4b