Question

Factorise: $ 32{x^4} - 500x $

Answer 1

Hint: In this question a polynomial is given and we have to factorise the given polynomial by using the grouping method and find the factors of the polynomial. Since the highest power of this polynomial is 4 therefore the number of factors should also be 4.

Complete step-by-step answer:
Given:
The polynomial given is –
 $ 32{x^4} - 500x $
Now there are only two terms in this polynomial, so using the grouping method, we find the common coefficients between both the terms.
The term $ 4x $ is common between both the terms so by taking $ 4x $ common, we can write this polynomial as –
$\Rightarrow 32{x^4} - 500x = 4x\left( {8{x^3} - 125} \right) $
Now to simplify this equation we can write –
$\Rightarrow 32{x^4} - 500x = 4x\left\{ {{{\left( {2x} \right)}^3} - {{\left( 5 \right)}^3}} \right\} $
Because the cube of 2 is 8 and the cube of 5 is 125.
So, the equation reduces in the formula form given below –
 $ {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} - ab + {b^2}} \right) $
Then the value of term $ \left\{ {{{\left( {2x} \right)}^3} - {{\left( 5 \right)}^3}} \right\} $ by using this formula becomes –
$\Rightarrow 32{x^4} - 500x = 4x\left\{ {\left( {2x - 5} \right)\left( {4{x^2} + 10x + 25} \right)} \right\} $
Now we see that the term $ \left( {4{x^2} + 10x + 25} \right) $ is also another polynomial having the highest power of the variable x as 2 so the number of the factors for this polynomial would be 2.
So, the total factors for the given polynomial $ 32{x^4} - 500x $ would be 4 and the simplest form of these factors can be given as –
 $ 32{x^4} - 500x = 4x\left( {2x - 5} \right)\left( {4{x^2} + 10x + 25} \right) $

Note: If we try finding the factors for the polynomial $ \left( {4{x^2} + 10x + 25} \right) $ by using the grouping method, it’s not possible to calculate because we cannot break the middle term $ 10x $ into two parts having coefficients common with the first and last term, therefore the factorization of this polynomial using the grouping method is not possible. But we know that if factors are calculated for a polynomial having the highest power of x as 2, we would get 2 factors, therefore we can say that this polynomial has 2 factors but we cannot calculate them by using the grouping method.