Question

Find the product of the following binomials\[\left( {x + 5} \right)\left( {y + 3} \right)\].

Answer 1

Hint: In the given question, we have to find the product of the given binomials. In order to find out the product, we will take the help of the distributive law of multiplication of binomials. Under the distributive law of binomials, each term of one bracket is multiplied with every term of the other bracket. After multiplication, we will simplify the result to find the ultimate solution.

Complete step-by-step solution:
Let us first understand what are polynomials, binomials, and distributive law that will help us in solving the question.
Simply put, polynomials are mathematical expressions consisting of variables and coefficients. Polynomials involve operations of multiplication, addition, and subtraction. Polynomials with just one variable are called monomials. For example: \[{x^2} + 3x + 7\]
Likewise, polynomials with two variables are known as binomials. For example: \[3x + 4xy + 6\].
The distributive law in mathematics is an operation of multiplication and addition. Under the distributive law, each term of one bracket or parentheses is multiplied with every other term of every other of the parentheses. For example: \[k(p + q) = kp + kq\]i.e., the monomial factor k is multiplied or applied to every other factor of the binomial\[(p + q)\].
Now, let us begin with the question,
Given binomials are, \[\left( {x + 5} \right)\left( {y + 3} \right)\]
To multiply the given binomials, we will multiply the first term of the first polynomial with each term of the second polynomial and then multiply the second term of the first polynomial with each term of the second polynomial.
Therefore, we get,
\[ \Rightarrow x\left( {y + 3} \right) + 5\left( {y + 3} \right)\]
\[ \Rightarrow xy + 3x + 5y + 15\] Which is our required product.

Note: It is important to note here that multiplication of polynomials is just about the distributive law. But one should be careful about the signs while multiplying as the multiplication of two negative terms gives a positive term. And multiplication of one negative and one positive term gives a negative term. Likewise, the multiplication of two positive gives positive results.