Multiplying Binomials Calculator (MBC) is a free online tool that multiplicates two binomials resulting in a trinomial expression. CalCon offers an online tool for calculating the product of two binomials, and you can also find the same as the CalCon Android or iOS application. The MBC performs the calculation fairly quickly and displays the trinomial or trinomial coefficients. An overview of all calculation steps is also available. The following text is a brief overview of the basic concepts related to binomials and polynomials.
Binomials
In math (algebra), a binomial is a polynomial representing the sum of two terms, each representing a monomial. It is the simplest type of polynomial after a monomial. Therefore, it is a polynomial that is the sum of two monomials. The binomial can be presented in general form as follows:
ax + bPolynomials
A polynomial consists of monomials that contain natural degrees of variables. Here is an example of a polynomial:
4x^{3} + 2x^{2} - 3 x +1Each part of a polynomial that is added is called a monomial. The example above is a polynomial with four monomials. The numbers that appear at the beginning of each variable are called coefficients. A number that appears alone without a variable is called a constant.
In this case, the coefficient with x3 is 4, the coefficient with x2 is 2, the coefficient with x is -3, and the constant is 1.
Each monomial in a polynomial has its degree. The degree of a monomial is the degree of a variable in that monomial.
- 4x3 has a level 3 and is called a cubic monomial.
- 2x2 has a power of 2 and is called a square monomial.
- -3x has degree 1 and is called a linear monomial.
- 1 has degree 0 and is called a constant.
By definition, the degree of a polynomial is equal to the greatest degree of its monomial. The above example is a polynomial of degree 3, called a cubic polynomial.
Polynomials can contain more than one variable. Here is another example of a polynomial:
t^{4} - 6s^{3}t^{2} -12st + 4s^{4} - 5This is a polynomial because all exponents of variables are natural numbers. Therefore, this polynomial contains five monomials.
In math, the degree of a monomial is equal to the sum of the degrees of all the variables in the monomial. In other words, the degree of a monomial is the number of variables that multiply in that monomial, regardless of whether they are the same or not.
- t4 has degree 4.
- -6s3t2 has degree 5.
- -12st has a grade of 2.
- 4s4 has degree 4.
- -5 is a constant, so the degree is 0.
Since the greatest degree of a monomial in this polynomial is 5, this is a polynomial of the fifth degree.
Polynomials are expressions that contain variables only in whole negative potentials. In other words, a polynomial cannot use variables within roots, logarithms, trigonometric functions, or any other modern math tool. However, they can consist of many variables.
As we mentioned earlier, a binomial is a polynomial with two members. The MBC deals only with linear binomials, i.e., multiplication of expressions of the form ax + b, which means with only one variable x in the first degree. Such binomials can often be found in learning materials, applications, and textbooks and are more than enough to explain the basic terms, concept and essence.
The result of multiplying two binomials using the MBC is a trinomial. In elementary algebra, a trinomial is a polynomial consisting of three terms; in other words, trinomial is the sum of three monomials, that is, the product of two binomials.
How to perform binomials multiplication operations?
When you need to multiply a binomial (or any polynomial, in the general case), the basic rule is that each member of the first expression is multiplied by each member of the second expression.
In algebra, a binomial is a two-term expression. It is, therefore, an algebraic expression with two members which can be added, subtracted, multiplied, and divided like all other arithmetic operations.
When multiplying two binomials, the first member of the first binomial should be multiplied by the first and second members of the second binomial. Accordingly, the second member of the first binomial should be multiplied by the first and second members of the second binomial. Thus, multiplication can be done four times when two binomials are multiplied.
Multiplication of two binomials can be performed by different methods, such as horizontal method, vertical method, FOIL method, etc. However, the FOIL method is used when multiplying two binomials. This means F-first, O-outer, I-inner, L-last.
Without going into too much detail, we’ll take two binomials: a₁x + a₀ and b₁x + b₀. Then, if we apply the mentioned rule, we get:
(a₁x + a₀) \times (b₁x + b₀) = (a₁x \times b₁x) + (a₁x \times b₀) + (a₀ \times b₁x) + (a₀ \times b₀)In the following, we will solve the expression using the math rules of commutativity and associativity.
(a₁x + a₀) \times (b₁x + b₀) = (a₁x \times b₁x) + (a₁x \times b₀) + (a₀ \times b₁x) + (a₀ \times b₀) = (a₁ \times b₁) \times x² + (a₁ \times b₀) \times x + (a₀ \times b₁) \times x + (a₀ \times b₀) (a₁x + a₀) \times (b₁x + b₀) = (a₁ \times b₁) \times x² + (a₁ \times b₀) \times x + (a₀ \times b₁) \times x + (a₀ \times b₀) = (a₁ \times b₁) \times x² + (a₁ \times b₀ + a₀ \times b₁) \times x + (a₀ \times b₀)The last expression represents the solution for multiplying binomials using FOIL method. Let us mention here only that it is not too difficult to move from binomial to polynomial in general. To multiply polynomials, there is only a bit more work left to do.
Multiply two binomials – an example
The multiplication operation is quite simple, and the following steps need to be done:
- Enter the coefficients of the first binomial in fields a1 and a0
- Enter the coefficients of the second binomial in fields b1 and b0
- See the result that is filled in automatically in fields c2, c1, and c0
- See the solutions step by step
In our case, the first expression is:
(2x+5)The second expression is:
(9x+6)And the resulting trinomial has coefficients of 18, 57, and 30. Solutions can be represented as:
18x^{2}+57x+30See also our Quadratic Formula Calculator.
