1: Chapters
- Page ID
- 45641
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.4: Fractions
- We can perform arithmetic operations with rational numbers (fractions). The two types of fractions we will encounter are called proper and improper: Proper fractions have value less than 1, for example 2/5 and 1/8. Observe that for these fractions the numerator is less than the denominator. Improper fractions have value greater than or equal to 1, for example 7/6 and 3/2. For these fractions the numerator is greater than the denominator.
- 1.6: Evaluating Expressions
- A mathematical expression that consists of variables, numbers and algebraic operations is called an algebraic expression. Each algebraic expression can contain several terms. For example, the expression above contains two terms: 30x and 20y. The numerical factor of each term is called a coefficient. The coefficients of the terms above are 30 and 20, respectively. When considering a variable term, we see that it is composed of a numerical coefficient and a variable part.
- 1.8: Scientific Notation
- To write 1 trillion (1 followed by 12 zeros) or 1 googol (1 followed by 100 zeroes) takes a lot of space and time. There is a mathematical scientific notation which is very useful for writing very big and very small numbers.
- 1.9: Polynomials
- A polynomial is a sum of monomials. A polynomial with one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. It has three terms
- 1.10: Adding and Subtracting Polynomial Expressions
- Polynomials can be added, subtracted, multiplied and divided. When adding or subtracting, we can only combine terms that are like terms.
- 1.11: Multiplying Polynomial Expressions
- In this chapter, we multiply polynomials. For the multiplication of a monomial by a polynomial, we need to distribute the monomial to multiply each term of the polynomial.
- 1.12: Dividing Polynomials
- In the previous chapter, we added, subtracted, and multiplied polynomials. Now, what remains is dividing polynomials. We will only consider division of a polynomial by a monomial.
- 1.13: Simplifying Square Roots
- Finding a square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself.
- 1.14: Factoring a Monomial from a Polynomial and GCF
- Factoring is extremely useful when we try to solve polynomial equations and simplify algebraic fractions. In the following three chapters, we will learn several methods of factoring.
- 1.15: Factoring the Difference of Two Squares
- In this chapter, we will learn how to factor a binomial that is a difference of two perfect squares.
- 1.19: Solving Linear Equations, Decimals, Rationals
- In this chapter we look at certain types of linear equations, those including decimal coefficients or rational coefficients. The reason why we discuss these separately is because we can “get rid” of the decimal numbers or denominators in the equation by performing a simple trick.
- 1.20: Word Problems for Linear Equations
- Word problems are important applications of linear equations. We start with examples of translating an English sentence or phrase into an algebraic expression.
- 1.21: Rewriting Formulas
- The language of mathematics is powerful. It is a language which has the ability to express relationships and principles precisely and succinctly. Faraday was a brilliant scientist who made history-making discoveries yet they were not truly appreciated until Maxwell was able to translate them into a workable language, that of mathematics.
- 1.22: Solving Quadratic Equations by Factoring
- As mentioned earlier, solving equations is dependent on the type of equation at hand. You can review solving linear equations in chapter 16. This chapter will deal with solving quadratic equations. These are equations that contain the second power of a variable and nothing higher.
- 1.23: Linear Inequalities
- In this section we solve linear inequalities.
- 1.30: Solving a System of Equations Graphically
- Since we know that graphs of linear equations are lines, it is natural to graph the lines representing our system and observe where they are with respect to one another in the coordinate plane. There are only three possible configurations: the lines intersect at a single point, the lines coincide, the lines are parallel. Finding this intersection (if possible) amounts to solving the linear system.