In this resource, we have compiled all the formulas related to algebra. Algebra is a field of mathematics in which we study mathematical symbols and those rules that manipulate these symbols. These symbols in elementary algebra reflect the quantities that have no fixed values. These quantities are known as variables. The relationship between different variables in algebra is depicted through equations. The arithmetic operations of addition, subtraction, multiplication, and division can be employed to solve complex problems in mathematics.

Monomials

A monomial is an algebraic expression that consists of a single term only. For instance, the expressions 2xy, 6yz, and 7 x are monomials because they contain one term only. Monomials are constructed by multiplying variables and whole numbers together. A constant or a number itself is a monomial. For example, the number 45987, 23, 40, and 9 are monomials. A single variable is also a monomial. For instance, p, q, x, y, and z are monomials. There are two rules of a monomial:

  • When we multiply two monomials together, we get a new monomial.
  • When a monomial is multiplied by a constant, the resulting expression is also a monomial.

Now, let us see what are some of the common monomial formulas:

ax^n + bc^n = (a + b)x^n

ax^n \cdot bx^m = (a \cdot b)x^{n + m}

ax^n : bx^m = (a : b) x ^ {n - m}

(ax^n)^m = a^m \cdot x ^ {n \cdot m}

 

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Binomials

Unlike a monomial, a binomial expression contains two terms. Any expression that has more than one term is also known as a polynomial. Hence, we can say that a binomial is a polynomial expression. A binomial is a combination of variables and constants. For instance, 3x + 5, 7xy - 9, and 2z + 1 are examples of binomial expressions. The terms in a binomial expression are segregated by the arithmetic operations of addition or subtraction. To be a binomial, an expression should fulfill the following conditions:

  • It must have two terms
  • If the variables of the two terms are the same, then their exponents (powers) must be different
  • The exponents of the variables must be positive integers. They should not be fractions or negative integers.

One rule of a binomial expression is that the exponents of the same variables in a single term should be different. What does it mean? Let suppose, we have the following expression with two terms that have the same variables and their exponents.

x ^2 + x^2

Since the exponents and variables are the same, hence we can easily add the above terms and the resulting answer will be monomial.

x ^2 + x^2 = 2x^2

So, to avoid this, the exponents of the same variables should be different in binomial expressions. For instance, consider the following expressions having two same variables, but different exponents:

x^2 + x^3

We cannot add them together. So, it is a binomial.

Some of the common binomial formulas are given below:

(a \pm b)^2 = a^2 \pm 2 \cdot a \cdot b + b^2

(a + b) \cdot (a - b) = a^2 - b^2

(a \pm b)^3 = a^3 \pm 3 \cdot a^2 \cdot b + 3 \cdot a \cdot b^2 \pm b^3

a^3 + b^3 = (a + b) \cdot (a^2 - ab + b^2)

a^3 - b^3 = (a - b) \cdot (a^2 + ab + b^2)

(x + a) (x + b) = x^2 + (a + b)x + ab

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Binomial Coefficient Formula

The general notation of the binomial coefficient is \binom {n}{k}. The formula for computing the binomial coefficient is given below:

 

\binom{n}{k} = \frac {n!} {k! (n-k!)}, where n \geq k \geq 0

It can be read as "n over k", or sometimes "n choose k".

It depicts the number of ways we can select unordered k number of outcomes from number of possibilities "n". The other formulas of binomial coefficient are:

\binom {n}{0} = \binom{n}{n} = 1

\binom{n}{k} = \binom {n}{n - k}

 

Trinomials

A trinomial is a polynomial expression that consists of three terms. These three terms are a combination of variables, constants, and coefficients multiplied together. Some examples of trinomial expressions are given below:

4x + 3y - 5

7xy + 8z - 20

7c + 2b = a

The formulas of trinomials are given below:

(a + b + c)^2 = a^2 + b^2 + c^2 + 2 \cdot a \cdot b + 2 \cdot a \cdot c + 2 \cdot b \cdot c

ax^2 + bx + c = 0

a \cdot (x - x_1) \cdot (x - x_2) = 0

 

Quadratic Formula

To find a solution to a quadratic equation, we use the quadratic formula. The quadratic equation is an equation that has the highest power 2. The general form of quadratic equation is given below:

ax^2 + bx + c = 0,    a \neq 0

The formula to solve the above quadratic equation is given below:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

If ax^2 = 0, then x = 0.

If x = 0, then ax^2 + bx = 0

x(ax + b) = 0

x = - \frac{b}{a}

Matrix Formulas

Matrix is a set of numbers that are arranged into definite numbers of rows and columns. A matrix can be of an order 2x2, 3x3, 4x4, or so on. The matrix which has equal number of rows and columns is known as square matrix.

A matrix can have unequal number of rows and columns. This type of matrix is known as a rectangular matrix.

The common formulas of the matrices are given below:

\begin {bmatrix} a_1 & a_2  \\ a_3 & a_4\\ \end {bmatrix} + \begin {bmatrix} b_1 & b_2  \\ b_3 & b_4\\ \end {bmatrix} = \begin {bmatrix} a_1 + b_ 1& a_2 + b_ 2  \\ a_3 + b_3 & a_4 + b_ 4\\ \end {bmatrix}

\begin {bmatrix} a_1 & a_2  \\ a_3 & a_4\\ \end {bmatrix} - \begin {bmatrix} b_1 & b_2  \\ b_3 & b_4\\ \end {bmatrix} = \begin {bmatrix} a_1 - b_ 1& a_2 - b_ 2  \\ a_3 - b_3 & a_4 - b_ 4\\ \end {bmatrix}

\lemda \cdot \begin {bmatrix} a_1 & a_2  \\ a_3 & a_4\\ \end {bmatrix} = \begin {bmatrix} \lemda \cdot a_1 & \lemda \cdot a_2  \\ \lemda \cdot a_3 & \lemda \cdot a_4\\ \end {bmatrix}

 

Matrix Formulas

If a matrix of order m x n is multiplied by the matrix of order n x p, then the new matrix will be of the order m x p.
\begin {bmatrix} a_1 & a_2  \\ a_3 & a_4\\ \end {bmatrix} \cdot \begin {bmatrix} b_1 & b_2  & b_3\\ b_4 & b_5 & b_6\\ \end {bmatrix} = \begin {bmatrix} a_1 \cdot b_1 + a_2 \cdot b_4 & a_1 \cdot b_2 + a_2 \cdot b_5 & a_1 \cdot b_3 + a _2 \cdot b_6\\ a_2 \cdot b_1 + a_4 \cdot b_4 & a_3 \cdot b_2 + a_ 4 \cdot b_5 & a_3 \cdot b_3 + a_4 \cdot b_6\\ \end {bmatrix}

Inverse of a Matrix

The formula for computing the inverse of the matrix is given below:

A ^ {-1} = \frac{1}{|A|} \cdot (A^*)^t

Here, A ^ {-1} means inverse of the matrix A.

|A| means determinants of the matrix A

A^* Adjugate or cofactors of the matrix A

(A^*)^t reflects transpose of the adjugate of matrix

 

When a matrix is multiplied by its inverse, then we get an identity matrix.

A \cdot A ^ {-1} = A ^ {-1} \cdot A = I

(A \cdot B) ^ {-1} = B^ {-1} \cdot A ^ {-1}

(A ^ {-1})^{-1} = A

(k \cdot A )^ {-1} = k^ {-1} \cdot A ^ {-1}

Determinants Formulas

Determinant of Order One

A matrix of order 1 has 1 row and 1 column in it. The determinant of the matrix of order 1 is the matrix itself:

|a_{11}| = a_ {11}

Determinant of Order Two

The determinant of the matrix of order two is calculated using the following formula:

\begin {vmatrix} a_1 & a_2  \\ a_3 & a_4\\ \end {vmatrix} = a_1a_3 - a_2a_4

Determinant of Order Three

The determinant of the matrix of order 3 is computed using the following formula:

B = \begin {bmatrix} a & b & c  \\ d & e & f \\ g & h & i \\ \end {bmatrix}

The determinant of the matrix B will be calculated as:

|B| = a \cdot\begin {vmatrix} e & f \\ h & i\end {vmatrix} - b \cdot \begin {vmatrix} d & f \\ g & i\end {vmatrix} + c \begin {bmatrix} d & e \\ g & h\end {bmatrix}

Rule of Sarrus

Rule of Sarrus is also called basketweave method. This rule presents an alternative way to compute the determinant of a 3x3 matrix.

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.