Question

How do you factor \[64{x^3} + 27\]?

Answer 1

Hint: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.