Question

Find out the product of the following polynomials \[\left( {q + 6} \right)\left( {q + 5} \right)\].

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 start solving our question
Given binomial expression is,
\[ \Rightarrow \left( {q + 6} \right)\left( {q + 5} \right)\]
Now, applying the law of distribution, we will multiply each term of the first bracket with each and every term of the second bracket.
Thus, we get
\[ \Rightarrow q\left( {q + 5} \right) + 6\left( {q + 5} \right)\]
Therefore,
\[ \Rightarrow {q^2} + 6q + 5q + 30\]
\[ \Rightarrow {q^2} + 11q + 30\] which is our required product.

Thus the answer is \[{q^2} + 11q + 30\].

Note: For the given question, one needs to remember exponents and their properties. An exponent refers to the number of times a number is multiplied by itself. For example, \[{4^5} = 4 \times 4 \times 4 \times 4 \times 4\], for this expression 4 is known as its base and 5 is the exponent. It is an important property of exponent that while multiplying, exponents with the same base can be added. Like we did for the given question where, \[\left( {{q^1}} \right)\left( {{q^{^1}}} \right) = {q^{1 + 1}} = {q^2}\].