Question

How do you factor $ {x^3} + 3{x^2} - 6x - 8 $ ?

Answer 1

Hint: Use the splitting of terms into pairs method to easily get the results. Just split the values into two parts first is $ ({x^3} - 8) $ (since we know, it can further be splitted) and second is $ (3{x^2} - 6x) $ (since it has some common factors to take). Just take the two parts and solve it separately and join the results at last to get the results.

Complete step-by-step answer:
Let $ {x^3} + 3{x^2} - 6x - 8 $ be represented as $ f(x) $ .
 $ f(x) = $ $ {x^3} + 3{x^2} - 6x - 8 $
Split the values of $ f(x) $ in two pairs.
 $ f(x) = $ $ ({x^3} - 8) + (3{x^2} - 6x) $
 Solve the paired terms separately:
Let’s take $ ({x^3} - 8) $ .It can be written as $ ({x^3} - {(2)^3}) $
Using the formula of $ ({a^3} - {b^3}) = (a - b)({a^2} + ab + {b^2}) $ solve the terms.
 $ ({x^3} - {(2)^3}) = (x - 2)({x^2} + 2x + {2^2}) = (x - 2)({x^2} + 2x + 4) $ …………………(i)
Let’s take $ (3{x^2} - 6x) $
Taking $ 3x $ common, we get:
 $ (3{x^2} - 6x) = 3x(x - 2) $ ………………….(ii)
Adding (i) and (ii) to get $ f(x) $ :
 $ f(x) = $ $ ({x^3} - 8) + (3{x^2} - 6x) $
 $ f(x) = $ $ (x - 2)({x^2} + 2x + 4) $ $ + 3x(x - 2) $
Taking $ (x - 2) $ common, we get:
 $ f(x) = $ $ (x - 2) $ $ ({x^2} + 2x + 4 + 3x) $
 $ = (x - 2) $ $ ({x^2} + 5x + 4) $
 $ f(x) = $ $ (x - 2) $ $ ({x^2} + 5x + 4) $ ……………………..(iii)
Further solving $ ({x^2} + 5x + 4) $ , using middle term factorization as:
 $
  ({x^2} + 5x + 4) = ({x^2} + 4x + x + 4) \\
   = ({x^2} + 4x) + (x + 4) \;
  $
Taking $ x $ common from $ ({x^2} + 4x) $ ,we get:
 $ ({x^2} + 5x + 4) = x({x^{}} + 4) + (x + 4) $
Taking $ (x + 4) $ common, we get:
\[({x^2} + 5x + 4) = ({x^{}} + 4)(x + 1)\]
Therefore, \[({x^2} + 5x + 4) = ({x^{}} + 4)(x + 1)\]
Put the values in eq(iii), we get:
 $ f(x) = $ \[(x - 2)({x^{}} + 4)(x + 1)\]
Hence, the factors of $ {x^3} + 3{x^2} - 6x - 8 $ are $ (x - 2),(x + 1) $ and $ (x + 4) $ .
Or, factors are $ f(2),f( - 1) $ and $ f( - 4) $
So, the correct answer is “ $ (x - 2),(x + 1) $ and $ (x + 4) $ ”.

Note: You can also use hit and trial method for the same question. In which Select a random no. put it in $ f(x) $ and check. If it gives $ 0 $ as output, then it is one of the factors of $ f(x) $ then simply divide $ f(x) $ with the factor to get the results.
First check whether the terms can be splitted or not, is there any formula for further simplification or not, then use the splitting term method. If there is no scope of further division in splitting by taking common or using a formula then use the hit and trial to simplify.