Question

How do you multiply $\left( {x - 2} \right)\left( {x + 3} \right)\left( {x - 4} \right)$?

Answer 1

Hint: Here, in the given question, we are given polynomials and we need to multiply them. As we can see each polynomial contains two terms it means we are given three binomials (a polynomial which contains two terms is called a binomial) and we need to multiply them. We will multiply given binomials using the distributive law of multiplication. According to this law, when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers and multiplied by each of them separately, then adding the two products together for the same result as multiplying the first number by the sum. First, we will multiply the first two binomials and then we will multiply the product of these binomials by the third binomial.

Complete step-by-step answer:
Now, let us write the expression given in question.
$\left( {x - 2} \right)\left( {x + 3} \right)\left( {x - 4} \right)$
Now, we will multiply the first two binomials using the distributive law of multiplication.
$\left( {x - 2} \right)\left( {x + 3} \right)$
$ \Rightarrow x \times \left( {x + 3} \right) - 2 \times \left( {x + 3} \right)$
Solve further by opening the brackets:
$ \Rightarrow {x^2} + 3x - 2x - 6$
Solve the like terms in the above expression:
$ \Rightarrow {x^2} + x - 6$
Now, we will multiply the above written expression with binomial $\left( {x - 4} \right)$.
$ \Rightarrow \left( {{x^2} + x - 6} \right)\left( {x - 4} \right)$
$ \Leftrightarrow \left( {x - 4} \right)\left( {{x^2} + x - 6} \right)$
$ \Rightarrow x \times \left( {{x^2} + x - 6} \right) - 4\left( {{x^2} + x - 6} \right)$
Solve further by opening the brackets:
$ \Rightarrow {x^3} + {x^2} - 6x - 4{x^2} - 4x + 24$
Solve the like terms in the above expression:
$ \Rightarrow {x^3} - 3{x^2} - 10x + 24$
Finally, we got all the like terms. This is the final answer.
So, the correct answer is “$ {x^3} - 3{x^2} - 10x + 24$”.

Note: While multiplying like bases, keep the base same and add the exponents. Remember that the product of two negative numbers is always positive and the product of one negative and one positive number is negative. Any two polynomials can be multiplied with each other by distributing each term individually.