Flammarion Engraving: A CC License

This subsection introduces algebraic simplification as a fundamental skill. It highlights that while individual algebraic techniques are relatively easy to grasp, the true challenge lies in applying the correct techniques within the broader context of problem-solving. The text emphasizes that students should practice recognizing when to use specific skills. An example is given illustrating the simplification of an expression using the distributive property. The importance of identifying and factoring out common factors, even after an initial simplification, is also stressed. The example demonstrates the need to repeatedly check for common factors to achieve complete simplification, going beyond simply finding the greatest common factor in one step.

This subsection focuses on factoring, a crucial algebraic technique. It begins by presenting a problem involving factoring a three-term expression, 42x²y⁶ + 98xy³ − 210x³y². The initial step involves finding the greatest common factor (GCF) of the coefficients and variables. The example guides the reader through the process of systematically identifying and factoring out the GCF. The concept of trinomial factoring is explained using examples such as factoring x² + 7x + 10. The sign of the constant term is identified as a key element in determining the approach, focusing on whether factors should add up to or have a difference equal to the middle term's coefficient. More complex trinomial factoring, involving a leading coefficient greater than 1, is also explored, showcasing the need to consider the interaction between the factors of the constant and the leading coefficient. A methodical approach, involving systematic exploration of factor pairs of the constant term, is recommended to solve these more complex problems.

This section provides a historical overview of solving quadratic equations, tracing their solution methods back thousands of years to Babylonian geometry. It mentions the contributions of Indian mathematician Brahmagupta, who used 'rhetorical algebra' in the 7th century, and Arab mathematicians of the 9th and 10th centuries. The work of Leonardo of Pisa (Fibonacci), who included Arab methods in his 1202 book 'Liber Abaci,' is also noted. The section then explores the question of why the quadratic formula works, explaining that it involves isolating the variable 'x', which is more complex than in linear equations because of the x² term. The process of 'completing the square' is mentioned as a useful technique. The text also discusses programming the quadratic formula into a graphing calculator, specifically mentioning the TI-84 series, illustrating a practical application of the formula and introducing basic programming concepts. The use of parentheses in the formula to ensure correct calculation is also emphasized, along with instructions for handling cases resulting in complex-valued answers.

A significant portion of the section addresses the complexities of solving quadratic equations that yield complex numbers. The text highlights that using the quadratic formula might result in the square root of a negative number, which was initially considered impossible until the development of complex numbers. The section explains how to adjust calculator settings (specifically for TI calculators) to allow for complex-valued answers. This involves navigating the calculator's mode settings to choose 'a+bi' instead of 'REAL,' enabling the calculator to compute and display complex solutions of the form a ± bi, where 'i' represents the imaginary unit (√-1). The implications of complex number solutions are discussed, although a deep dive into complex number theory isn't presented at this stage.

This subsection focuses on the mechanics of multiplying and dividing rational expressions. It emphasizes that simplifying rational expressions by canceling common factors before performing the multiplication or division is often easier than simplifying afterward. The process is directly analogous to multiplying or dividing numerical fractions: multiply numerators together and denominators together. The key takeaway is that any factor in any numerator can be canceled with any factor in any denominator before performing the main calculation. This method streamlines the process, making it less error-prone and improving efficiency.

This subsection details three main methods for solving rational equations: multiplying both sides of the equation to eliminate denominators, cross-multiplying, and creating common denominators. The text emphasizes that while distinct in approach, these methods all stem from the same underlying principle. The choice of method is often a matter of preference or convenience depending on the particular equation. The section does not delve deeply into the specific application of each method but rather establishes the three core techniques available for tackling rational equations. It positions these methods as a natural extension of the techniques developed for manipulating rational expressions.

This section builds upon previous sections on solving equations by graphing, extending the techniques to solve polynomial inequalities. Instead of simply finding the x-values where y = 0 (as in solving equations), the focus shifts to identifying the range of x-values for which y is either greater than or less than 0, depending on the inequality. This involves analyzing the graph of the polynomial function to determine the intervals where the function's value satisfies the given inequality. The approach is presented as a natural progression from solving equations graphically, leveraging the graphical representation to determine the solution sets of polynomial inequalities.

This subsection extends the graphical method to solve rational inequalities, which involve ratios of polynomials (fractions). The key difference here is the introduction of asymptotes—lines that the graph approaches but never touches. The presence of asymptotes impacts how the solution intervals are determined. The method involves analyzing the graph of the rational function to find the intervals where the function's value satisfies the given inequality, taking into account the behavior of the function near the asymptotes. The section highlights that solving rational inequalities graphically requires careful consideration of the asymptotes' influence on the function's behavior and the resulting solution intervals. The term 'asymptote,' derived from a Greek root meaning 'not meeting,' is explicitly defined within the context of rational function graphs.

This section begins by differentiating between additive, multiplicative, and polynomial relationships, providing examples of each. It then introduces exponential relationships, contrasting them with other types of relationships. The section highlights that while exponential growth can model some natural phenomena, real-world growth is typically constrained by resource limitations. This leads to a discussion of logistic functions, which better represent situations with limited resources. The concept of the mathematical constant e (approximately 2.71828) is explained within the context of exponential growth and its area under the curve. The section includes a worked example involving the application of an exponential function in a real-world scenario, specifically tracking the amount of medication in a patient's bloodstream over time. Both graphical and algebraic methods are mentioned for solving such problems.

This section introduces logarithmic notation and its relationship to exponential relationships. It provides a historical context, mentioning John Napier's work in the early 1600s. Although the computational usefulness of logarithms has diminished with the advent of calculators, the section emphasizes the enduring conceptual importance of logarithms in mathematics. The core definition of a logarithm is presented: logb N = x is equivalent to bx = N. The focus here is on understanding and becoming familiar with logarithmic notation as a restatement of exponential relationships. The section aims to establish a foundational understanding of logarithmic notation, setting the stage for its application in solving equations in later sections.