Question

Multiply the binomials (a + 3b) and (x + 5).
(a) (a + 3b) + (x + 5)
(b) (a + 3b) (x + 5)
(c) (a + 3b) – (x + 5)
(d) 5 (a + 3b)

Answer 1

Hint: Understand the term ‘binomial’ and check each option one – by – one. Eliminate the wrong options to get the correct one. Use the information: - ‘+’ sign indicates addition of terms and ‘-’ sign indicates subtraction of terms. If there is no sign between two terms enclosed in brackets then it is assumed to be the multiplication operation.

Complete step-by-step answer:
Here, we have been given two binomial terms: - (a + 3b) and (x + 5), we have to find the expression for their multiplication among the given four options. First, let us see the definition of the term ‘binomial’.
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. A binomial can be represented as: -
\[A{{x}^{m}}-B{{x}^{n}}\], where A and b are numbers, m and n are non – negative integers and x is called as a variable.
Now, let us come to the question. We have been provided with two binomials (a + 3b) and (x + 5). These binomials can be written as: -
\[\begin{align}
  & \Rightarrow \left( a+3b \right)=\left( a{{x}^{0}}+3b{{x}^{0}} \right) \\
 & \Rightarrow \left( x+5 \right)=\left( {{x}^{1}}+5{{x}^{0}} \right) \\
\end{align}\]
As we can see that exponents of the variable x are non – negative, therefore they are called binomials.
Now, to find the multiplication expression of these binomials, let us check each option one – by – one.
(a) (a + 3b) + (x + 5)
Here, ‘+’ sign is in between the two binomials and we know that ‘+’ sign indicates addition of the terms. So, option (a) is not correct.
(b) (a + 3b) (x + 5)
Here, no sign is in between the two terms and we know that when two expressions are enclosed in brackets and there is no sign between them, then it represents multiplication. So, option (b) is correct.
(c) (a + 3b) – (x + 5)
Here, ‘-’ sign is in between the two binomials and we know that ‘-’ sign indicates subtraction of the terms. So, option (c) is not correct.
(d) 5 (a + 3b)
Here, we have 5 multiplied with (a + 3b) but 5 is not a binomial but it is a monomial. So, here we have a monomial multiplied with a binomial. So, option (d) is not correct.
Hence, the conclusion is that option (b) is the correct answer.

So, the correct answer is “Option (b)”.

Note: One must know the signs used for different arithmetic operations like ‘+’ for addition, ‘-’ for subtraction, ‘x’ for multiplication and ‘\[\div \]’ for division. But sometimes for multiplication we will not be provided with any sign. It will just be written as two or more terms enclosed in brackets like \[\left( x-a \right)\left( x-b \right)\left( x-c \right).....\]. Sometimes multiplication is represented by a dot (.) sign. So, we must remember all the signs to solve the questions.