Question

How do you factor $64{x^3} + 27$?

Answer 1

Here in this question, we have to find the factors of the given equation. If you see the equation it is in the form of ${a^3} + {b^3}$. We have a standard formula on this algebraic equation and it is given by ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$, hence by substituting the value of a and b we find the factors.

Complete step by step explanation:
The equation is an algebraic equation or expression, where algebraic expression is a combination of variables and constant.

Now consider the given equation $64{x^3} + 27$, let we write in the exponential form. The
number 64 can be written as $4 \times 4 \times 4$ and the $64{x^3}$can be written as $4x \times 4x \times 4x$, in the exponential form it is ${\left( {4x} \right)^3}$.
The number 27 written as $3 \times 3 \times 3$ and in exponential form is ${3^3}$.

Therefore, the given equation is written as ${\left( {4x} \right)^3} + {3^3}$, the equation is in the form of ${a^3} + {b^3}$.${a^3} + {b^3}$.

We have a standard formula on this algebraic equation and it is given by ${a^3} + {b^3} = (a + b)({a^2} - ab + {b^2})$, here the value of a is $4x$ and the value of b is 3.

By substituting these values in the formula, we have
$64{x^3} + 27 = {\left( {4x} \right)^3} + {3^3} = (4x + 3)({(4x)^2} - (4x)(3) + {3^2})$

On simplifying we have
$\Rightarrow 64{x^3} + 27 = (4x + 3)(16{x^2} - 12x + 9)$

The second term of the above equation can be solved further by using factorisation or by using the formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Let we consider $16{x^2} - 12x + 9$, and find factors for this. Here a=16, b=-12 and c=9. By
substituting these values in the formula we get
$x = \dfrac{{ - ( - 12) \pm \sqrt {{{( - 12)}^2} - 4(16)(9)} }}{{2(16)}}$

On simplification we have
$\Rightarrow x = \dfrac{{12 \pm \sqrt {144 - 576} }}{{32}}$
$\Rightarrow x = \dfrac{{12 \pm \sqrt { - 432} }}{{32}}$

On further simplifying we get an imaginary number so let us keep as it is.

Therefore, the factors of $64{x^3} + 27$ is $(4x + 3)(16{x^2} - 12x + 9)$

Note: To find the factors for algebraic equations or expressions, it depends on the degree of the equation. If the equation contains a square then we have two factors. If the equation contains a cube then we have three factors. Here this equation also contains 3 factors, the two factors are imaginary.