Question

Find the product $24{x^2}(1 - 2x)$ and evaluate it for $x = 2$.

Answer 1

Hint: The given question asks us to find the product and evaluate an expression at a given value. If $y = f(x)$ is an expression and we are asked to evaluate it at a value of $x = a$, then it means that we have to find the value of $f(a)$ such that $a$ replaces $x$ in the expression $f(x)$.

Complete step by step solution:
The given expression in the question is $24{x^2}(1 - 2x)$. This expression is itself a combination of two more expressions, i.e. $24{x^2}$ and $1 - 2x$. To find the product of $24{x^2}(1 - 2x)$, we will multiply $24{x^2}$ and $1 - 2x$, such that
$ \Rightarrow 24{x^2}(1 - 2x) = 24{x^2} \times (1 - 2x)$
We will now use the distributive property of multiplication over subtraction. For three numbers $a$, $b$ and $c$, this property is stated as $a(b - c) = ab - ac$.
For $24{x^2}(1 - 2x)$, we have $a = 24{x^2}$, $b = 1$ and $c = 2x$. On substituting these values in the property, we will get
$ \Rightarrow 24{x^2}(1 - 2x) = 24{x^2} \times (1 - 2x)$
$ \Rightarrow 24{x^2}(1 - 2x) = 24{x^2} \times 1 - 24{x^2} \times 2x$
On further multiplying the right hand side of the above equation, we will get
$ \Rightarrow 24{x^2}(1 - 2x) = 24{x^2} - 48{x^3}$
Hence, the product of $24{x^2}(1 - 2x)$ is $24{x^2} - 48{x^3}$.
We will now evaluate $24{x^2} - 48{x^3}$ for $x = 2$. This means we will put the value of $x$ as $2$ in the expression $24{x^2} - 48{x^3}$, such that it becomes
$ = 24{(2)^2} - 48{(2)^3}$
$ = 24(2 \times 2) - 48(2 \times 2 \times 2)$
$ = 24 \times 4 - 48 \times 8$
$ = 96 - 384$
$ = 96 - 384$
$ = - 288$

Hence, when we evaluate $24{x^2} - 48{x^3}$ for $x = 2$, we get $ - 288$ as the answer.

Note:
It must be kept in mind that the given question is an expression and not an equation. Also, the product obtained $24{x^2} - 48{x^3}$ is a cubic expression as the highest power of the variable $x$ is $3$. We can also say that the given expression is of degree $3$.