Pre-Calculus

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How to Decompose Partial Fractions

A process called partial fractions takes one fraction and expresses it as the sum or difference of two other fractions. In calculus, this process is useful before you integrate a function. Because integration [more…]

How to Factor a Polynomial Expression

In mathematics, factorization or factoring is the breaking apart of a polynomial into a product of other smaller polynomials. If you choose, you could then multiply these factors together, and you should [more…]

How to Solve Linear Systems

When you solve systems with two variables and therefore two equations, the equations can be linear or nonlinear. Linear systems are usually expressed in the form Ax + By [more…]

How to Solve Systems that Have More than Two Equations

Larger systems of linear equations involve more than two equations that go along with more than two variables. These larger systems can be written in the form Ax + By + Cz + . . . [more…]

Even-Odd Identities in Trigonometric Functions

All functions, including trig functions, can be described as being even, odd, or neither. Knowing whether a trig function is even or odd can help you simplify an expression. These even-odd identities are [more…]

Completing the Square for Conic Sections

When the equation of a conic section isn't written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows: [more…]

Finding the Key Parts of All Hyperbolas

A hyperbola is the set of all points in the plane such that the difference of the distances from two fixed points (the foci) is a positive constant. Hyperbolas always come in two parts, and each one is [more…]

Rules for Adding and Subtracting Matrices

To add or subtract matrices, you have to operate on their corresponding elements. In other words, you add or subtract the first row/first column in one matrix to or from the exact same element in another [more…]

Logarithm Basics

Logarithms are simply another way to write exponents. Exponential and logarithmic functions are inverses of each other. For solving and graphing logarithmic functions [more…]

How to Graph Polynomials

Although it may seem daunting, graphing polynomials is a pretty straightforward process. Once you have found the zeros for a polynomial, you can follow a few simple steps to graph it. [more…]

How to Find a Greatest Common Factor in a Polynomial

No matter how many terms a polynomial has, you always want to check for a greatest common factor (GCF) first. If the polynomial has a GCF, factoring the rest of the polynomial is much easier because once [more…]

How to Use the FOIL Method to Factor a Trinomial

For polynomials with a nonprime leading coefficient and constant term, you can use a procedure called the FOIL method of factoring (sometimes called the [more…]

How to Factor a Perfect Square

FOIL stands for multiply the first, outside, inside, and last terms together. When you FOIL a binomial times itself, the product is called a perfect square. [more…]

How to Factor a Difference of Squares

When you FOIL (multiply the first, outside, inside, and last terms together) a binomial and its conjugate, the product is called a difference of squares. [more…]

How to Break Down a Cubic Difference or Sum

After you’ve checked to see if there’s a Greatest Common Factor (GCF) in a given polynomial and discovered it’s a binomial that isn’t a difference of squares, you should consider that it may be a difference [more…]

Factoring Four or More Terms by Grouping

When a polynomial has four or more terms, the easiest way to factor it is to use grouping. In this method, you look at only two terms at a time to see if any techniques become apparent. For example, you [more…]

Finding the Roots of a Factored Equation

In pre-calculus, you can use the zero-product property to find the roots of a factored equation. After you factor a polynomial into its different pieces, you can set each piece equal to zero to solve for [more…]

How to Solve a Quadratic Equation When It Won’t Factor

When asked to solve a quadratic equation that you just can’t seem to factor (or that just doesn’t factor), you have to employ other ways of solving the equation, such as by using the quadratic formula. [more…]

How to Complete the Square

Completing the square comes in handy when you’re asked to solve an unfactorable quadratic equation and when you need to graph conic sections (circles, ellipses, parabolas, and hyperbolas). [more…]

How to Find the Real Roots of a Polynomial Using Descartes’s Rule of Signs

If you know how many total roots a polynomial has, you can use a pretty cool theorem called Descartes’s rule of signsto count how many roots are real numbers [more…]

How to Find Imaginary Roots Using the Fundamental Theorem of Algebra

The fundamental theorem of algebra can help you find imaginary roots. Imaginary roots appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign [more…]

Basics of How to Guess and Check Real Roots

You can use the rational root theorem to narrow down the search for roots of polynomials. While Descartes’s rule of signs only narrows down the real roots into positive and negative, the rational root [more…]

How to Guess and Check Real Roots — 1 — List All Possible Rational Roots

When you look for all the possible rational roots of any polynomial, the first step is to use the rational root theorem to list them all.

The rational root theorem says that if you take all the factors [more…]

How to Guess and Check Real Roots — 2 — Testing Roots by Dividing Polynomials Using Long Division

Once you have used the rational root theorem to list all the possible rational roots of any polynomial, the next step is to test the roots. One way is to use long division of polynomials and hope that [more…]

How to Guess and Check Real Roots — 3 — Testing Roots by Dividing Polynomials Using Synthetic Division

Once you have used the rational root theorem to list all the possible rational roots of any polynomial, the next step is to test the roots. One way is to use synthetic division. Synthetic division is a [more…]

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