Discriminant Notes
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The document discusses the quadratic formula and the discriminant. The discriminant determines the nature of the roots of a quadratic equation. A positive discriminant that is a perfect square yields two real roots, a positive non-perfect square discriminant yields two real irrational roots, a zero discriminant yields one real rational root (a double root), and a negative discriminant yields two complex (imaginary) roots. Several examples of applying the discriminant to determine the root types are provided.
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- Euclid's five common understandings (axioms) and two postulates regarding the existence of lines and planes through given points.
Rational Root Theorem
The Rational Root Theorem provides a method to find all possible rational roots of a polynomial with integer coefficients. It states that every rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Examples are provided to demonstrate finding all possible rational roots using this theorem and then checking them to determine the actual rational roots. Once a rational root is found, synthetic division can be used to find the depressed polynomial which can then be fully factored to obtain all factors of the original polynomial.
Synthetic Division Notes
This document provides instruction on the method of synthetic division. It explains that synthetic division can be used to divide polynomials when the divisor is a first-degree (linear) polynomial. The steps of synthetic division are outlined, which involve writing the dividend in expanded standard polynomial form, then systematically "dropping" and multiplying coefficients to solve for the quotient polynomial. Examples are provided to demonstrate the process.
Intro to Polynomials
This document discusses polynomials and their graphs. It defines a polynomial as an expression with two or more algebraic terms separated by plus and minus signs. Polynomials are named based on their degree and number of terms, with the degree being the highest exponent of its terms. The general shapes of polynomial graphs are constant for linear polynomials, parabolic for quadratics, cubic for cubics, and so on. Important features of polynomial graphs discussed include x-intercepts, y-intercepts, regions of increase/decrease, relative maxima/minima, and end behavior determined by degree and leading coefficient.
Square Root Function Transformation Notes
This document provides information and examples for graphing square root functions, including:
- Tips for graphing square root functions such as how the h and k terms translate the function and how the a term forms the shape.
- Examples of square root functions with questions asking to identify the minimum/maximum point, domain, range, x-intercepts and y-intercepts.
- Explanations of how to find the minimum/maximum point from a graph or equation, noting it is the (h,k) point.
- A description of vertex form for square root functions where the a term reflects or dilates, and h and k translate, with the minimum/maximum point again being
Simplify Nth Roots Notes
To simplify nth roots, the radicand must have no perfect nth power factors. This can be done through prime factorization or taking out the largest perfect root factor. Practice problems are provided to simplify nth roots with numbers, variables, and combinations using these methods so the exponent of any variable in the radicand and the index of the radical have a greatest common factor of one.
Roots and Radical Expressions Notes
This document provides information on radicals and root expressions. It defines the key parts of a radical expression, including the index and radicand. It explains that roots and radicals are inverse operations of exponents, and how to "undo" a power or radical. Examples are given of various nth roots and their relationships to exponents. The document outlines the rules for principal roots, noting even roots have two solutions while odd roots only have one. It cautions the reader to check the index when evaluating roots. Finally, it discusses how to simplify nth roots involving variables by dividing the exponent by the index.
Rational Exponents
This document discusses rational exponents and how they can be used to rewrite radicals. It provides the general rule that the denominator of a rational exponent indicates the index of the radical, while the numerator indicates the exponent of the radicand. It then lists some properties of rational exponents, such as how to simplify expressions using operations like multiplication, division, powers, and negative exponents. Examples are provided to demonstrate how to use these properties to rewrite expressions and simplify radicals.
Operations with Radicals
This document discusses operations with radicals. It covers multiplying radicals by combining like terms, dividing radicals by dividing coefficients and radicands when possible and rationalizing denominators to remove radicals, and FOILing with radicals by using the rules for multiplying and adding/subtracting. Examples are provided for simplifying expressions with radicals and for rationalizing denominators when dividing radicals.
Dividing Radicals using Conjugates
This document provides instructions for dividing radicals and rationalizing denominators. It discusses simplifying coefficients and radicands when dividing radicals. When rationalizing denominators, it explains multiplying the numerator and denominator by a radical term called the conjugate to eliminate radicals in the denominator. The conjugate is formed by changing the sign of the second term of a binomial. Examples are provided to demonstrate rationalizing denominators using conjugates.
Quadratic Transformations Notes
This document provides notes and instructions for graphing quadratic functions and transformations. It includes tips for graphing quadratics using squares, reflections, and dilations. Examples of quadratic functions are worked through step-by-step to find the vertex, domain, range, x-intercepts, and y-intercepts. Formulas are provided for writing quadratic functions in vertex form to identify the reflection/dilation, horizontal translation, and vertical translation values.
Graphing Quadratic Functions in Standard Form
This document discusses graphing quadratic functions of the form y = ax^2 + bx + c. It provides the following key points:
- Quadratic functions produce parabolic graphs that open up or down depending on whether a is positive or negative.
- The vertex of the parabola is the point of minimum or maximum, which corresponds to the line of symmetry that passes through it.
- To graph a quadratic, one finds the line of symmetry, determines the vertex coordinates, and plots at least four other points to connect into a smooth curve.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
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Chapter 6
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3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
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Discriminant Notes
- 1. Quadratic Formula & The Discriminant Algebra 2
- 2. What is the discriminant? 𝑥 = −𝑏 ± 𝑏2 − 4𝑎𝑐 2𝑎
- 3. Why do we want to know the value of the discriminant? Positive, perfect square– 2 Real, rational roots Positive, non-perfect square– 2 Real, irrational Zero – 1 Real, Rational (double root) Negative – 2 Complex (imaginary) roots
- 4. Examples: 9𝑥2 − 3𝑥 − 8 = −10 −2𝑥2 − 8𝑥 − 14 = −6
- 5. Examples: 2𝑥2 + 5𝑥 − 4 = 0 −7𝑥2 + 16𝑥 = 8𝑥
- 6. Examples: −6𝑥2 + 7𝑥 + 3 = 0 2𝑥2 − 10𝑥 − 5 = 0
- 7. Examples: Find the value of a such that 𝑎𝑥2 + 3𝑥 + 5 = 0 has two real roots (solutions).
- 8. Examples: Find the value of a such that 𝑎𝑥2 + 48𝑥 + 64 = 0 has one real root (solution).
- 9. Examples: Find the value of a such that 𝑎𝑥2 + 3𝑥 − 6 = 0 has two imaginary roots (solutions).
- 10. Examples: Find the value of c such that 2𝑥2 − 6𝑥 + 𝑐 = 0 has two imaginary roots (solutions).
- 11. Examples: Find the value of c such that −4𝑥2 + 8𝑥 + 𝑐 = 0 has two real roots (solutions).