The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are provided to find the sum of roots one at a time as -b/a, two at a time as c/a, three at a time as -d/a, and so on. An example is also given to demonstrate using the formulas.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
- The trigonometric ratios of sine, cosine, and tangent can be used to find the measures of angles in right triangles.
- For a given angle, the trig ratios are defined as the ratio of the length of the side opposite to the angle over the length of the hypotenuse (sine), the ratio of the length of the side adjacent to the angle over the hypotenuse (cosine), and the ratio of the opposite side over the adjacent side (tangent).
- The inverse trig functions (cotangent, secant, cosecant) are defined in terms of the tangent, cosine, and sine ratios respectively.
This document discusses polynomial functions of higher degree and their applications in modeling. It covers graphs of polynomial functions, end behavior, zeros, the intermediate value theorem, and modeling. Key topics include transforming monomial graphs, finding extrema and zeros, applying the leading term test to determine end behavior, and using factorization to sketch graphs. The document provides examples and explanations of these polynomial concepts.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression.
3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.
1) Henri Becquerel discovered that uranium salts would expose photographic plates even when wrapped in black paper, showing they emitted invisible "rays" he called radioactivity.
2) Marie Curie discovered the radioactive elements polonium and radium, and found radium was over a million times more radioactive than uranium.
3) Ernest Rutherford discovered there were at least two types of radiation, which he called alpha and beta based on how far they could penetrate matter and their opposite electric charges.
This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
1) Angles can be measured in degrees, minutes, or radians. Trigonometric functions relate to the sides of a right triangle and depend on the angle of rotation.
2) Positive angles are measured clockwise from the positive x-axis, negative angles counterclockwise.
3) The value of a trig function for any angle can be determined using a calculator, right triangles, or trig identities involving reference angles.
- The trigonometric ratios of sine, cosine, and tangent can be used to find the measures of angles in right triangles.
- For a given angle, the trig ratios are defined as the ratio of the length of the side opposite to the angle over the length of the hypotenuse (sine), the ratio of the length of the side adjacent to the angle over the hypotenuse (cosine), and the ratio of the opposite side over the adjacent side (tangent).
- The inverse trig functions (cotangent, secant, cosecant) are defined in terms of the tangent, cosine, and sine ratios respectively.
This document discusses polynomial functions of higher degree and their applications in modeling. It covers graphs of polynomial functions, end behavior, zeros, the intermediate value theorem, and modeling. Key topics include transforming monomial graphs, finding extrema and zeros, applying the leading term test to determine end behavior, and using factorization to sketch graphs. The document provides examples and explanations of these polynomial concepts.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression.
3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.
1) Henri Becquerel discovered that uranium salts would expose photographic plates even when wrapped in black paper, showing they emitted invisible "rays" he called radioactivity.
2) Marie Curie discovered the radioactive elements polonium and radium, and found radium was over a million times more radioactive than uranium.
3) Ernest Rutherford discovered there were at least two types of radiation, which he called alpha and beta based on how far they could penetrate matter and their opposite electric charges.
This document introduces methods for solving quadratic equations beyond factoring, including the square root property, completing the square, and the quadratic formula. It discusses how to determine the number and type of solutions based on the discriminant. The key steps are presented for solving quadratics, graphing quadratic functions as parabolas, and finding the domain and range. Piecewise-defined quadratic functions are also explained.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths in different triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
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.
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.
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)Nigel Simmons
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
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.
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.
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/
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.
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.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths in different triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
The document outlines the cosine rule for finding the length of the side of a triangle opposite an angle when the other two sides and included angle are known. It derives the formula c^2 = a^2 + b^2 - 2abcosC and provides examples of applying the rule to solve for side lengths of triangles.
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.
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.
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)Nigel Simmons
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
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.
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.
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/
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.
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.
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.
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.
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
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
5. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
6. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b
a
7. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
8. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
9. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
Quartics ax 4 bx 3 cx 2 dx e 0
10. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
Quartics ax 4 bx 3 cx 2 dx e 0
b
a
11. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
Quartics ax 4 bx 3 cx 2 dx e 0
b c
a a
12. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
Quartics ax 4 bx 3 cx 2 dx e 0
b c
a a
d
a
13. Roots and Coefficients
Quadratics ax 2 bx c 0
b c
a a
Cubics ax 3 bx 2 cx d 0
b c
a a
d
a
Quartics ax 4 bx 3 cx 2 dx e 0
b c
a a
d e
a a
15. For the polynomial equation;
ax n bx n1 cx n2 dx n3 0
b
a (sum of roots, one at a time)
16. For the polynomial equation;
ax n bx n1 cx n2 dx n3 0
b
a (sum of roots, one at a time)
c
a (sum of roots, two at a time)
17. For the polynomial equation;
ax n bx n1 cx n2 dx n3 0
b
a (sum of roots, one at a time)
c
a (sum of roots, two at a time)
d
a (sum of roots, three at a time)
18. For the polynomial equation;
ax n bx n1 cx n2 dx n3 0
b
a
(sum of roots, one at a time)
c
a
(sum of roots, two at a time)
d
a
(sum of roots, three at a time)
e
a
(sum of roots, four at a time)
19. For the polynomial equation;
ax n bx n1 cx n2 dx n3 0
b
a
(sum of roots, one at a time)
c
a
(sum of roots, two at a time)
d
a
(sum of roots, three at a time)
e
a
(sum of roots, four at a time)
Note:
2
2 2
20. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
21. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5
2
22. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3
2 2
23. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
24. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
25. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
26. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b)
27. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b)
28. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b)
3
2
1
2
29. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b)
3
2
1
2
3
30. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b) c) 2 2 2
3
2
1
2
3
31. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the
values of;
a) 4 4 4 7
5 3 1
2 2 2
5 1
4 4 4 7 4 7
2 2
27
2
1 1 1
b) c) 2 2 2
2
2
3
2
1
2
3
34. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
35. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
36. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
37. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
1
38. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
1
1 1 1
(iii)
39. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
1
1 1 1
(iii)
1 1 1
40. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
1
1 1 1
(iii)
1 1 1
3
1
41. 1988 Extension 1 HSC Q2c)
If , and are the roots of x 3 x 1 0 find:
3
(i)
0
(ii)
1
1 1 1
(iii)
1 1 1
3
1
3
42. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
43. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
44. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
2
45. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
2
3
46. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
P 3 0
2
3
47. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
P 3 0
2
3 2 3 3 k 3 6 0
3 2
48. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
P 3 0
2
3 2 3 3 k 3 6 0
3 2
54 9 3k 6 0
49. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
P 3 0
2
3 2 3 3 k 3 6 0
3 2
54 9 3k 6 0
3k 39
50. 2003 Extension 1 HSC Q4c)
It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 are
reciprocals of each other.
Find the value of k.
1
Let the roots be , and
1 6
P 3 0
2
3 2 3 3 k 3 6 0
3 2
54 9 3k 6 0
3k 39
k 13
51. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
52. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
53. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
54. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
55. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
1 1 s
56. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
1 1 s
s 2
57. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
1 1 s 1 t
s 2
58. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
1 1 s 1 t
s 2 t 2
59. 2006 Extension 1 HSC Q4a)
The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real
numbers, has three real zeros, 1, and
(i) Find the value of r
1 r
r 1
(ii) Find the value of s + t
1 1 s 1 t
s 2 t 2
s t 0