Calculus

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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…]

How to Eliminate Exponents from Trigonometric Functions Using Power-Reducing Formulas

Power-reducing formulas allow you to get rid of exponents on trig functions so you can solve for an angle's measure. This ability comes in very handy in calculus. [more…]

How to Prove an Equality Using Reciprocal Identities

Oftentimes, your math teachers will ask you to prove equalities that involve the secant, cosecant, or cotangent functions. Whenever you see these functions in a proof, the reciprocal identities usually [more…]

How to Simplify an Expression Using Even/Odd Identities

Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. Sine and tangent are both odd functions, and cosine is an even function. In other [more…]

How to Simplify an Expression Using Co-function Identities

If you take the graph of y = sin x and shift it to the left by pi/2 units, it looks exactly like the graph of y= cos x. The same is true for tangent and cotangent, as well as secant and cosecant. That's [more…]

How to Prove an Equality by Using Periodicity Identities

Using the periodicity identities comes in handy when you need to prove an equality that includes the expression (x + 2pi) or the addition (or subtraction) of the period. For example, to prove [more…]

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