Question

How do you multiply the 3 binomials.

Answer 1

Hint: Here in this question, we have to find the product of 3 binomials. The binomial is one form of algebraic expression. So let us 3 binomials which are different from one another and then we use the arithmetic operation that is multiplication and then we simplify.

Complete step-by-step answer:
The binomial concept will come under the topic of algebraic expressions. The algebraic expression is a combination of variables and constant. The alphabets are known as variables and the numerals are known as constants. In algebraic expression or equation, we have 3 types namely, monomial, binomial and polynomial.
A polynomial equation with two terms joined by the arithmetic operation + or – is called a binomial equation.
Now let us consider the three binomial and they are
$ (x + 3) $ , $ (x + 2) $ and $ (x + 5) $
Now we have to multiply the binomials, to multiply the binomials we use multiplication. The multiplication is one of the arithmetic operations.
Now we multiply the above 3 binomials we get
 $ (x + 3) \times (x + 2) \times (x + 5) $
Here dot represents the multiplication. First, we multiply the first two terms then we obtain the product and to that resultant product we again multiply by the binomial.
 $ \Rightarrow \left( {x(x + 2) + 3(x + 2)} \right)(x + 5) $
On multiplying we get
 $
   \Rightarrow \left( {{x^2} + 2x + 3x + 6} \right)(x + 5) \\
   \Rightarrow \left( {{x^2} + 5x + 6} \right)\left( {x + 5} \right) \;
  $
By multiplying the two terms we have obtained the product, again we multiply the result to the binomial we get
 $
   \Rightarrow \left( {{x^2} + 5x + 6} \right)\left( {x + 5} \right) \\
   \Rightarrow {x^2}(x + 5) + 5x(x + 5) + 6(x + 5) \\
   \Rightarrow {x^3} + 5{x^2} + 5{x^2} + 25x + 6x + 30 \;
  $
On simplifying we get
 $ \Rightarrow {x^3} + 10{x^2} + 31x + 30 $
Hence, we have multiplied the three binomials and obtained the trinomial expression or equation. The product is also an algebraic expression.

Note: To multiply we use operation multiplication, multiplication of numbers is different from the multiplication of algebraic expression. In the algebraic expression it involves the both number that is constant and variables. Variables are also multiplied, if the variable is the same then the result will be in the form of an exponent.