EPPS Math and Coding Camp

Algebra review

Instructor: Azharul Islam

2.1 BASIC PROPERTIES OF ARITHMETIC

  • Associative properties: \((a+b)+c = a+(b+c)\) and \((a \times b) \times c = a \times (b \times c)\).
  • Commutative properties: \(a+b=b+1\) and \(a \times b=b \times a\).
  • Distributive property: \(a(b+c)=ab+ac\).
  • Identity properties state that there exists a zero such that \(x+0=x\) and that there exists a one such that \(x \times 1 =x\).
  • Inverse property states that there exists a \(-x\) such that \((-x)+x=0\). An inverse under multiplication, \(x^{-1}\), such that \((x^{-1} \times x) = 1\).

2.1.1 Order of Operations

parentheses, exponents, multiplication, division, addition, subtraction

A common mnemonic device people use to memorize order of operations is

PEMDAS

or Please Excuse My Dear Aunt Sally.

2.1.2 Ratios, Proportions, and Percentages

  • The ratio of two quantities is one divided by the other \(\frac{x}{y}\) is the ratio of x to y. Ratios are also written as \(x:y\).
  • Though a ratio may be negative, we typically consider ratio variables that range from \(0\) to \(\infty\).
  • The proportion of two variables, on the other hand, is the amount one vari able represents of the sum of itself and a second variable: \(|\frac{x}{x+y}|\). A proportion ranges from a minimum of \(0\) to a maximum of \(1\).
  • The proportion of expenditures that is spent in a given category is often interested.
  • The percentage one variable represents of a total is the proportion represented over the range from \(0\) to \(100\): \(|\frac{x}{x+y}| \times 100 \%\)
    Many people find a percentage representation more intuitive than a proportion representation.

You will also encounter the percentage change in a variable, which is calculated as: \(\frac{x_{t+1}-x_t}{x_t}\), where the subscript \(t\) indicates the first observation and the subscript t + 1 indicates the second observation.

2.2 ALGEBRA REVIEW

2.2.1 Fractions

\[ \frac{Numerator}{Denominator} \]

2.2.2 Factoring

  • Factoring involves rearranging the terms in an equation to make further manipulation possible or to reveal something of interest.

Example:
\[ \delta+\delta^2+4\delta-6\delta^2+18\delta^3\]

In this case we combine all the terms that have the same exponent, which gives us \[ 18\delta^3-5\delta^2+5\delta\]

2.2.2.1 Factoring Quadratic Polynomials

Quadratic polynomials are composed of a constant and a variable that is both squared and raised to the power of one: \(x^2-2x+3\), or \(7-12x+6x^2\). Quadratic polynomials can be factored into the product of two terms: \((x \pm ?) \times (x \pm ?)\) where you need to determine whether the sign is \(+\) or \(-\), and then replace the question marks with the proper values.

2.2.2.2 Factoring and Fractions

Consider the fraction \(\frac{x^2-1}{x-1}\)

We can factor the numerator \(x^2-1=(x+1)(x-1)\). We can thus rewrite the fraction as follows:

\[ \frac{x^2-1}{x-1}= \frac{(x+1)(x-1)}{x-1} \]

The term \(x-1\) is in both the numerator and the denominator and thus (as long as \(x\neq 1\)) cancels out, leaving \(x+1\) for \(x \neq 1\)

2.2.3 Expansion: The FOIL Method

\[ (\delta + \gamma)^2 \]

\[=(\delta + \gamma)(\delta + \gamma)\]

\[ =\delta^2 + 2\delta\gamma + \gamma^2 \]

F: Multiply the first terms: \((\underline{2\pi}+7) (\underline{4}+3\pi)=2 \pi \times 4=8 \pi\)

O: Multiply the outer terms: \((\underline{2\pi}+7) (4+\underline{3\pi})=2 \pi \times 3 \pi=6 \pi^2\)

I: Multiply the inner terms: \((2\pi+\underline{7}) (\underline{4}+3\pi)=4 \times 7=28\)

L: Multiply the last terms: \((2\pi+\underline{7}) (4+\underline{3\pi})=7 \times 3 \pi=21 \pi\)

Add terms to get: \(8\pi+6\pi^2+28+21\pi\)

Finally, group like terms to get: \(6\pi^2+29\pi+28\)

To test yourself, factor the final expression and show it yields the simplified expression with which we started. This is one way to check your work for any mistakes.

2.2.4.1 Solving Quadratics

Example:

\[ (x-3)^2=4 \]

\[ x-3=\pm 2 \]

\[ x=5 \text{ or } x=1 \]

Note that this quadratic equation will have two solutions in the real numbers, or zero, but not one. In other words, the cardinality of the solution set for a quadratic equation will be zero or two.

Example with no solutions in the real numbers: \(x^2+1=0\)

Another example: \[ x^2+8x+6=0 \]

The general form of a quadratic equation:
\[ ax^2+bx+c=0 \]

The general solutions to this equation: \[ x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \]

2.2.5 Inequalities

  • All pairs of real numbers have exactly one of the following relations: \(x=y\), \(x>y\), or \(x<y\).
  • Adding any number to each side of these relations will not change them; this includes the inequalities.
    That is, inequalities have the same addition and subtraction properties as equalities such that: if \(x>y\), then \(x+a>y+a\) and \(x-a>y-a\)
  • You flip the \(<\) or \(>\) sign when multiplying or dividing by a negative. Multiplying or dividing an inequality by zero is not allowed.
  • For multiplication, if \(a\) is positive and \(x>y\), then \(ax>ay\).
    If \(a\) is negative and \(x>y\), then \(ax<ay\).
  • For division, if \(a\) is positive and \(x>y\), then \(\frac{x}{a}>\frac{y}{a}\).
    If \(a\) is negative and \(x>y\), then \(\frac{x}{a}<\frac{y}{a}\).

Example:
Solve for \(y\):
\[ -4y>2x+12 \]

\[ y<-\frac{x}{2}-3 \]

Exercise:

\[ 1.4x^2+3.7x+1.1=0 \]

Answer:

\[ a=1.4, b=3.7, c=1.1 \]

\[ x=\frac{-3.7 \pm \sqrt{3.7^2-4 \times 1.4 \times 1.1}}{2.8} \]

Using a calculator to solve, we find that \(x=-.341 \text{ or } x=-2.301\)

Problems

Problem 1

Complete the following equations

  1. \(\prod_{m=6}^9 x_m = \ldots\)

  2. \(\sqrt[3]{27} = \ldots\)

Problem 2

Simplify this expression as much as possible: \(\frac{2x^2 + 20x + 50}{2x^2 - 50}\)

Problem 3

FOIL: \((2x - 3)(5x + 7)\)

Problem 4

Complete the square and solve for y: \(\frac{1}{3}y^2 + \frac{2}{3}y - 16 = 0\)

Problem 5

Solve using the quadratic formula: \(2x^2 + 5x - 7 = 0\)

Problem 6

Solve: \(-\delta > \frac{\delta+4}{7}\)

Problem 7

A firm faces the marginal revenue schedule \(MR = 80 - 2q\) and the marginal cost schedule \(MC = 15 + 0.5q\), where \(q\) is the quantity produced. You know that a firm maximizes profit when \(MC = MR\). What will the profit-maximizing output be?

Problem 8

Derive the quadratic formula by completing the square for the equation \(ax^2 + bx + c = 0\).

Any Questions?

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