Chapter F, Problem 1DE

To determine

To calculate: The factor of $x^3-27$.

Answer to Problem 1DE

Solution:

The factor of $x^3-27$ is $\left(x-3\right)\left(x^2+3x+9\right)$.

Explanation of Solution

Given Information:

The expression, $x^3-27$.

Formula used:

The difference of the two expressions that are cubes denoted by $A^3$ and $B^3$ is denoted by the formula;

$A^3-B^3=\left(A-B\right)\left(A^2+AB+B^2\right)$

Calculation:

Consider the expression, $x^3-27$.

As the difference of the two expressions that are cubes denoted by $A^3$ and $B^3$ is denoted by the formula;

$A^3-B^3=\left(A-B\right)\left(A^2+AB+B^2\right)$

Here, $x^3-27=x^3-3^3$

So,

$\begin{array}{l}A=x\\ B=3\end{array}$

Thus, put the values in $A^3-B^3=\left(A-B\right)\left(A^2+AB+B^2\right)$.

Thus, it becomes,

$\begin{array}{c}{x}^3-3^3=\left(x-3\right)\left(x^2+x\cdot 3+3^2\right)\\ =\left(x-3\right)\left(x^2+3x+9\right)\end{array}$

Hence, the factor of $x^3-27$ is $\left(x-3\right)\left(x^2+3x+9\right)$.