College Algebra Cheat Sheet

College algebra can be a source of constant frustration throughout your college life. It can earn you multiple Fs on tests, further complicating your academic life. And if you fear asking for help from your professor or friends, the situation aggravates.

Likewise, you have probably heard that algebra is the language of mathematics. But as it turns out, the language of algebra is also pretty easy to understand.

Additionally, there is a simple solution if you have tried all and failed. This is the college algebra cheat sheet. It is a list of the most important concepts in college algebra, covering everything from basic terms and properties of equality to graphing functions. With it, you will look forward to your algebra assignments and exams.

Want to learn further about this cheat sheet? Let’s jump right in.

College algebra cheat sheet

This is the college algebra cheat sheet.

Solving Polynomials

Solving polynomials is a fundamental part of algebra. The most basic method for solving polynomial equations is the substitution method. To solve the first-degree equation using this method, multiply both sides by the coefficient of one of the variables to isolate that variable on one side of the equation. 

Then substitute this value into the other variable’s term and simplify if necessary. Further, you may have to solve a polynomial when solving a quadratic equation or when you need to find the roots of a function. The process of solving polynomials can be broken down into two steps;

Factorization: This step involves separating all the factors that are in common between two or more terms in the polynomial (a factor is any number that divides evenly into another number).

Also see: Can mathway solve word problems?

For example, if we were trying to solve 5x+12x-10x-45, we would first factor out 5x from each term, leaving us with 5(x+2)(x-5).

Multiplication: After factoring out all the common factors, we multiply each factor in each term by its corresponding coefficient (the number next to each term).

So, for example, if we were solving 5(x+2)(x-5), we would multiply 5 times 1 from each term and then 2 times 2 from each term, and finally -5 times -5 from each term. The result of this process should leave us with a single term that contains only integers.

Systems of Linear Equations

A system of linear equations is a set of two or more equations written in the same format. Systems of linear equations are sets of linear equations that can be solved as a group using the same methods described below.

Solving systems of linear equations means finding values for all variables in an equation so that each equation in the system is true (or satisfied). Here’s how to solve a system:

• Write down all the variables and coefficients from each equation.
• Label each variable with its corresponding letter (x, y, z, etc.).
• Add like terms together, if possible, so every term has only one variable on its left side and one coefficient on its right side.
• Subtract any term from another term when they have like terms on both sides; this will leave you with one less term to deal with later on!
• Multiply or divide any term by another

Laws of Exponents

The Laws of Exponents are rules for raising a base to an exponent. These laws can simplify exponents, multiply, divide, and negate them.

Let’s take a look at each law individually

Law of Multiplication: If you multiply two numbers having the same base, the product will have that base raised to the power of both factors. For example, 5(5) = 52 = 25.

Law of Division: If you divide one number by another with the same base, the quotient will have that base raised to the power of both factors divided by their exponent. For example, 5÷5 = 5(1)= 5= 1/5.

Negative Exponent Rule: A negative exponent indicates how many times to subtract its coefficient from the original number before raising it to that power. For example, (-2)3 means -2×(-2)×(-2).

Factoring Polynomials

Factoring a polynomial is a fundamental concept in algebra. It can be done in several different ways. The first step is to look at the coefficients of the polynomial and determine the greatest common factor (GCF). Once you find the GCF, you should then use this to find any other factors that may be present in the polynomial. 

There is more than one way to factor a polynomial, but some methods are more efficient than others. The following section will cover some of these methods;

• Integer Factorization Method

This method is also known as trial division or long division method. This method begins by dividing the first term by all possible divisors of the second term until a zero remainder is found.

Once this happens, we know that there must be another factor besides 1 and 2 that divides both terms evenly into two parts. We do this for every possible divisor until we find one with no remainder, and this will be your answer.

• Factor Pair Method

This method uses trial division to factor each term separately before combining them into pairs whose product equals something similar to what we started with (products). This method works best when there is one common factor between both terms; however

Rules for Radicals and Fractional Exponents

A radical expression is a number, variable, or product raised to a power. For example, \(x^2\) is the square of x, \(x^3\) is the cube of x, and so on. A radical expression can be positive or negative depending on the value of x.

The radicand (the number under the radical sign) must not contain any fractions or exponents. The base (the number written outside the radical sign) can be a fraction or an exponent.

To simplify radicals, we use the following rules;

• Square root: If numbers are under the square root sign, multiply them and then take their square root.
• Cube root: If numbers are under the cube root sign, multiply them and then take their cube root.
• Fourth root: If there are numbers under the fourth root sign, then multiply them and then take their fourth root

Logarithm Rules

Logarithms are a way to simplify complicated calculations. They are used in calculus and many other math courses. Knowing how to use them is vital to avoid wasting time and make your life easier.

Logarithms are written using the base 10 (for example, the log of 5 is 1/2) or base e (for example, the log of 5 is 2). For example, if you have a number like 456 and want to find its log, write it as 456 = 50 × 10 + 6 × 1 + 5 × 0 + 4 × 1 + 3 × 10 + 2 × 100. This will give you: log(456) = 22/2 = 11/2 = 5½

If you asked for the base 10 logarithm of a number, just move the decimal place over two spaces. For example, if you have a number like 456 and want its base 10 logarithm, move the decimal over two spaces from the right side of the number

Quadratics

Quadratic is a type of function in algebra that includes the square function. Quadratic functions are important in many branches of mathematics and science, as they describe many phenomena in both the natural and social sciences. 

For example, they can be used to model the flight of an object under gravity or how water flows from a garden hose. A quadratic formula is a helpful tool for solving quadratic equations. It uses two solutions for each quadratic equation.

You’re almost halfway there if you know how to solve linear equations. The second step is eliminating the square root by dividing it into both sides. 

Simplification

Simplification is a term used in algebra to describe the process of changing a complicated expression into one that is easier to work with. For example, if you have the expression x + 2x + 3x, you can simplify this by dividing both sides by 2. 

The following is a list of standard simplification techniques;

• Simplify Rational Expressions – To simplify rational expressions, cancel out like terms and combine like terms.
• Simplify Algebraic Fractions – Add or subtract the numerators and denominators until they are equal (or as close as possible) to simplify algebraic fractions.
• Simplify Radical Expressions – To simplify radical expressions, use the properties of radicals to remove radicals from inside expressions (often using variations of the factoring method).
• Simplify Quadratic Equations – To simplify quadratic equations, factor them into two binomials (using various methods) and then solve each binomial separately (using multiple ways).

Final Thoughts

A solid understanding of college algebra is not only an excellent tool for future study and application, but it will help you develop skills in critical analysis and pattern recognition.

Now that we’ve covered most of the basics of college algebra, you should feel confident in taking your next test. Math is ultimately about problem-solving, and with a few tricks up your sleeve, you’ll be able to solve (almost) any problem. 

So don’t sweat it too much—just use our tips and tricks when you need them, and try to keep your mind focused while studying. After that, it’s all in your hands.