# Algebra

### Linear Algebra

#### Eigenspace (with Examples)

What is Eigenspace? Eigenspace is the span of a set of eigenvectors. These vectors correspond to one eigenvalue. So, an eigenspace always maps to a fixed eigenvalue. It is also a subspace of the original vector space. Finding it is equivalent to calculating ... more

#### Eigenvalues and Eigenvectors (Examples & Questions)

Eigenvalues and eigenvectors help solve varied problems. What are they, how do we find and use them? The vectors that are scaled by matrix multiplication are called eigenvectors of a matrix. The scale factor is called the eigenvalue. We can multiply a square ... more

### Polynomials

#### Solving Rational Equations (+ Questions)

The rational equations have the form Q(x)P(x)​=0, where P(x) and Q(x) are polynomials. The solution set of such an equation contains all the values of x which satisfy the equation. We can solve a rational equation by transforming it to a simple equation ... more

#### Factoring Cubic Polynomials (+Questions)

Factoring polynomials is necessary for solving many types of math problems. Sometimes it becomes challenging when we encounter a cubic polynomial. One way is to find the roots by applying the cubic formula, but it is too complex to remember and use. However, ... more

#### Factoring Polynomials (Methods + Questions)

Factoring polynomials can be challenging, particularly for degrees 3 (or more) and containing multiple variables. There are many ways to factorize them. We choose a method depending on the polynomial. List of Factorization Methods The polynomials can be ... more

#### Factoring a Quadratic in 4 Easy Steps (with Questions)

We encounter quadratic expressions and equations in a variety of problems in maths, physics, and engineering. In almost every case, we have to factorize it. The most common approach is to split the middle term. Another way is to find its root using the ... more

#### Finding Roots of Quadratic (Equation, Examples & Graphs)

The root of a polynomial p(x) is the value of x, such that p(x) = 0. So finding the root is equivalent to solving the equation p(x) = 0. Usually, finding the roots of a higher degree polynomial is difficult. Fortunately, for a quadratic, we have a simple ... more

#### Quadratic Function Graph (Examples & Graphing)

Some quadratic graphs are upwards, some downwards, some intersect with the x-axis, others not. What are the attributes of these graphs? How can we use them in graphing the function? Attributes of a Quadratic Graph The primary features of a quadratic graph are ... more

### Miscellaneous

#### Speed Formula (with Questions)

Speed is a measure of how quickly an object moves. It is the distance traveled by a body in unit time. For example, if a kid runs 50 meters in 10 seconds, we can say that he moves 5 meters every second. The unit time in this example is one second. So his ... more

#### Challenging Speed, Time, Distance Questions (Hints & Answers)

✩ You must already know: Speed, Time, Distance Formula : Speed=TimeDistance​ The Pythagoras Theorem Quadratic Equations Questions Try to use a minimum number of hints to reach the solution. All the best! If you find these challenging, learn to solve the ... more