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How do you simplify ${\left( {x - 3} \right)^3}$ ?

Last updated date: 20th Jun 2024
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Hint: In this question we need to find the simplified form of ${\left( {x - 3} \right)^3}$. Here, $\left( {x - 3} \right)$being multiplied $3$ times that is $\left( {x - 3} \right)\left( {x - 3} \right)\left( {x - 3} \right)$, the output of it will yield an equation that has ${x^3}$ as the variable with largest power, thus the equation will be cubic equation.

Complete step by step solution:
As we know that the equation of the form $a{x^3} + {x^2} + cx + d$ is known as the cubic equation. Here, $x$ is the variable, $a$, $b$, and $c$ are the coefficients and$d$ is the constant. The largest power of the variable in the equation is three and it determines that the equation is cubic.
In this question, we have given a term that is and we need to simplify it.
Here, is multiplied three times.
Here, we will consider the algebraic identity ${\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a + b} \right)$
Now we will apply this algebraic identity to the given equation as,
$\Rightarrow {\left( {x - 3} \right)^3} = {x^3} - {3^3} - 3x\left( 3 \right)\left( {x + 3} \right)$
Now, we will simplify the above expression as,
$\Rightarrow {\left( {x - 3} \right)^3} = {x^3} - 9 - 9x\left( {x + 3} \right)$
Now, we will multiply $9x$ in the bracket terms as,
$\Rightarrow {\left( {x - 3} \right)^3} = {x^3} - 9 - 9{x^2} - 27x$
After simplification we will get,
$\therefore {\left( {x - 3} \right)^3} = {x^3} - 9{x^2} - 27x - 9$

From above, the expanded form of the ${\left( {x - 3} \right)^3} = {x^3} - 9{x^2} - 27x - 9$.

Note:
The exponent is the number of times a number can be multiplied by itself. For example consider a variable $a$ as $a \times a = {a^2}$ , then ${a^n}$ represents $a$ multiplied by itself n number of times. The exponent form${a^n}$ is pronounced as a raise to the power n. Where,
${a^0} = 1$ And${a^1} = a$. The other properties are${\left( {ab} \right)^n} = {a^n}{b^n}$, also ${a^n}{a^m} = {a^{m + n}}$.