Question

How do you factor completely $ {x^4} + 5{x^2} - 36 $ ?

Answer 1

Hint: We use both grouping method and vanishing method to solve the problem. We take common terms out to form the multiplied forms. Factorising a polynomial by grouping is to find the pairs which on taking their common divisor out, give the same remaining number. In the case of vanishing method, we use the value of $ x $ which gives the polynomial value 0.

Complete step by step answer:
We apply the middle-term factoring or grouping to factorise the polynomial.
In case of $ {x^4} + 5{x^2} - 36 $ , we break the middle term $ 5{x^2} $ into two parts of $ 9{x^2} $ and $ - 4{x^2} $ .
So, $ {x^4} + 5{x^2} - 36 = {x^4} + 9{x^2} - 4{x^2} - 36 $ . We have one condition to check if the grouping is possible or not. If we order the individual elements of the polynomial according to their power of variables, then the multiple of end terms will be equal to the multiple of middle terms.
Here multiplication for both cases gives $ - 36{x^4} $ . The grouping will be done for $ {x^4} + 9{x^2} $ and $ - 4{x^2} - 36 $ .
We try to take the common numbers out.
For $ {x^4} + 9{x^2} $ , we take $ {x^2} $ and get $ {x^2}\left( {{x^2} + 9} \right) $ .
For $ - 4{x^2} - 36 $ , we take $ - 4 $ and get $ - 4\left( {{x^2} + 9} \right) $ .
The equation becomes $ {x^4} + 5{x^2} - 36 = {x^4} + 9{x^2} - 4{x^2} - 36 = {x^2}\left( {{x^2} + 9} \right) - 4\left( {{x^2} + 9} \right) $ .
Both the terms have $ \left( {{x^2} + 9} \right) $ in common. We take that term again and get
 $
  {x^4} + 5{x^2} - 36 \\
   = {x^2}\left( {{x^2} + 9} \right) - 4\left( {{x^2} + 9} \right) \\
   = \left( {{x^2} + 9} \right)\left( {{x^2} - 4} \right) \\
  $
Now we find the factorisation of the equation $ {x^2} - 4 $ using the identity of $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $ . We take $ {x^2} - 4 = {x^2} - {2^2} $ .
Therefore, we get
 $
  {x^2} - 4 \\
   = {x^2} - {2^2} \\
   = \left( {x + 2} \right)\left( {x - 2} \right) \\
  $
The final factored form will be $ {x^4} + 5{x^2} - 36 = \left( {{x^2} + 9} \right)\left( {x + 2} \right)\left( {x - 2} \right) $ .

Note:
 We find the value of x for which the function $ f\left( x \right) = {x^4} + 5{x^2} - 36 $ . We can see $ f\left( 2 \right) = {2^4} + 5{\left( 2 \right)^2} - 36 = 16 + 20 - 36 = 0 $ . So, the root of the $ f\left( x \right) = {x^4} + 5{x^2} - 36 $ will be the function $ \left( {x - 2} \right) $ . This means for $ x = a $ , if $ f\left( a \right) = 0 $ then $ \left( {x - a} \right) $ is a root of $ f\left( x \right) $ . We can also do the same process for $ \left( {x + 2} \right) $ .