Question

Find out the product of the given polynomials \[\left( {w + 4} \right)\left( {{w^2} + 3w - 6} \right)\].

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 every other bracket. After multiplication, we will add the powers and simplify it further to reach the ultimate product.

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 \[(p + q)\].

Let us start with the question,
Given expression,
\[ \Rightarrow \left( {w + 4} \right)\left( {{w^2} + 3w - 6} \right)\]
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,
\[ \Rightarrow \left( w \right)\left( {{w^2} + 3w - 6} \right) + \left( 4 \right)\left( {{w^2} + 3w - 6} \right)\]
\[ \Rightarrow {w^3} + 3{w^2} - 6w + 4{w^2} + 12w - 24\]
Now we will add the terms with the same base to simplify the expression further, \[ \Rightarrow {w^3} + 7{w^2} + 6w - 24\] which is our required product.

Thus the answer is \[{w^3} + 7{w^2} + 6w - 24\].

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, \[\left( {{w^1}} \right)\left( {{w^{^2}}} \right) = {w^{2 + 1}} = {w^3}\].