Question

Find the factors of \[27{x^3} + 64\].

Answer 1

Hint: When factoring the sum of cubes, the formula used is \[{a^3} + {b^3} = \left( {a + b} \right)({a^2} - ab + {b^2})\]. In this case \[27{x^3} + 64\] it can be observed that both the numbers have to be represented in the cubic form to find the factors. After doing the cubic representation the values are substituted in the formula \[{a^3} + {b^3} = \left( {a + b} \right)({a^2} - ab + {b^2})\].

Complete answer:
Since the equation given is in cubic form so the formula that can be used to find the factors of \[27{x^3} + 64\]is\[{a^3} + {b^3} = \left( {a + b} \right)({a^2} - ab + {b^2})\].
In this case of \[27{x^3} + 64\] we can assume,
\[
  27{x^3} = {a^3} \\
  64 = {b^3} \\
 \]
Now we can find a by solving,
\[27{x^3} = {a^3}\]
Taking cube root both the sides we have,
\[ \Rightarrow 3x = a\]
On solving b we have,
\[64 = {b^3}\]
Taking cube root both the sides we have,
\[4 = b\]
Now substituting \[a = 3x\]and \[b = 4\] in the equation \[{a^3} + {b^3} = \left( {a + b} \right)({a^2} - ab + {b^2})\]we have,
\[\left( {3x + 4} \right)\left( {{{\left( {3x} \right)}^2} - \left( {3x \times 4} \right) + {4^2}} \right)\]
On expanding the above equation we have,
\[ \Rightarrow \left( {3x + 4} \right)\left( {9{x^2} - 12x + 16} \right)\]
Hence, on factoring \[27{x^3} + 64\] we have the factored form as \[\left( {3x + 4} \right)\left( {9{x^2} - 12x + 16} \right)\].

Note: In general, to factorize a cubic polynomial, we find one factor by trial and error. Then the factor theorem or formulae are used to confirm that the guess is a root. Then the cubic polynomial can be divided by the factor to obtain a quadratic. Once we have the quadratic, we can apply the standard methods to factorize the quadratic equation. While solving or factoring any cubic polynomial it is important that all the numbers have to be represented in the cubic form. After cubic representation is once done then the values can be substituted in the formula. After the substitution it is important to evaluate so that exact factors can be obtained.