Calculus

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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…]

How to Use the Roots of a Polynomial to Find Its Factors

The factor theorem states that you can go back and forth between the roots of a polynomial and the factors of a polynomial. In other words, if you know one, you know the other. At times, your teacher or [more…]

How to Graph Polynomials When the Roots Are Imaginary Numbers — An Overview

In pre-calculus and in calculus, certain polynomial functions have non-real roots in addition to real roots (and some of the more complicated functions have [more…]

How to Solve an Exponential Equation with a Variable on One or Both Sides

Whether an exponential equation contains a variable on one or both sides, the type of equation you’re asked to solve determines the steps you take to solve it. [more…]

How to Solve an Exponential Equation by Taking the Log of Both Sides

Sometimes you just can’t express both sides of an exponential equation as powers of the same base. When facing that problem, you can make the exponent go away by taking the log of both sides. For example [more…]

How to Calculate the Sine of an Angle

Because you spend a ton of time in pre-calculus working with trigonometric functions, you need to understand ratios. One important ratio in right triangles is the sine. The [more…]

How to Calculate the Cosine of an Angle

Because you spend a ton of time in pre-calculus working with trigonometric functions, you need to understand ratios. One important ratio in right triangles is the cosine. The [more…]

How to Express Solutions for Inequalities with Interval Notation

You can use interval notation to express where a set of solutions begins and where it ends. Interval notation is a common way to express the solution set to an inequality, and it’s important because it’s [more…]

How to Verify the Inverse of a Function

At times, your textbook or teacher may ask you to verify that two given functions are actually inverses of each other. To do this, you need to show that both [more…]

Understanding the Properties and Identities of Logs

You need to know several properties of logs in order to solve equations that contain them. Each of these properties applies to any base, including the common and natural logs: [more…]

How to Solve Logarithmic Equations

Logarithmic equations take different forms. As a result, before solving equations that contain logs, you need to be familiar with the following four types of log equations: [more…]

How to Find the Sine of a Doubled Angle

You use a double-angle formula to find the trig value of twice an angle. Sometimes you know the original angle; sometimes you don't. Working with double-angle formulas comes in handy when you're given [more…]

How to Find the Tangent of a Doubled Angle

The double-angle formula for tangent is used less often than the double-angle formulas for sine or cosine; however, you shouldn't overlook it just because it isn't as popular as its cooler counterparts [more…]

How to Use Half-Angle Identities to Evaluate a Trig Function

You can use half-angle identities to evaluate a trig function of an angle that isn't on the unit circle by using one that is. For example, 15 degrees, which isn't on the unit circle, is half of 30 degrees [more…]

How to Express Products of Trigonometric Functions as Sums or Differences

If you can break up a product of trig functions into the sum of two different terms, each with its own trig function, doing the math becomes much easier. In pre-calculus, problems of this type usually [more…]

How to Express Sums or Differences of Trigonometric Functions as Products

It's a good idea to familiarize yourself with a set of formulas that change sums to products. Sum-to-product formulas are useful to help you find the sum of two trig values that aren't on the unit circle [more…]

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