Question

How do you factor $ {x^3} + 7{x^2} + 12 $ ?

Answer 1

Hint: To determine the factors of the above cubic equation ,first pull out $ x $ from each term as it is common in all .Now your function is split into linear and quadratic parts. To further factorise the quadratic part use the splitting up the middle term method to completely factorize the given function .

Complete step-by-step solution:
Given a Cubic equation $ {x^3} + 7{x^2} + 12 $ ,let it be $ f(x) $
 $ f(x) = {x^3} + 7{x^2} + 12x $
Pulling out $ x $ from each term as it is common in all, our equation becomes
 $ f(x) = x\left( {{x^2} + 7x + 12} \right) $
Now to factorise the quadratic part of the above expression we’ll be using splitting the middle term method by Comparing it with the standard Quadratic equation $ a{x^2} + bx + c $
a becomes 1
b becomes 7
And c becomes 12
So first calculate the product of coefficient of $ {x^2} $ and the constant term which comes to be
 $ = (12) \times 1 = 12 $
Now second Step is to the find the 2 factors of the number 2 such that the whether the addition or subtraction of those numbers is equal to the middle term or coefficient of x and the product of those factors results in the value of constant .
So if we factorize $ 12 $ , the answer comes to be 4 and 3 as $ 4 + 3 = 7 $ that is the middle term. and \[4 \times 3 = 12\] which is perfectly equal to the constant value.
Now writing the middle term sum of the factors obtained, so equation $ f(x) $ becomes
 $ f(x) = x\left( {{x^2} + 4x + 3x + 12} \right) $
Now taking common from the first 2 terms and last 2 terms
 $ f(x) = x\left( {x\left( {x + 4} \right) + 3\left( {x + 4} \right)} \right) $
Finding the common binomial parenthesis, the equation becomes
 $ f(x) = \left( x \right)\left( {x + 4} \right)\left( {x + 3} \right) $
Hence , We have successfully factorised our cubic equation $ f(x) $ as $ \left( x \right)\left( {x + 4} \right)\left( {x + 3} \right) $ .

Additional Information:
Cubic Equation: A cubic equation is a equation which can be represented in the form of $ a{x^3} + b{x^2}cx + d $ where $ x $ is the unknown variable and a,b,c,d are the numbers known where $ a \ne 0 $ .If $ a = 0 $ then the equation will become quadratic equation and will no more cubic. The degree of the quadratic equation is of the order 3.Every Cubic equation has 3 roots.

Note:
1. One must be careful while calculating the answer as calculation error may come.
2.Don’t forget to compare the given cubic equation with the standard one every time.
3. Solution to the cubic equation can be obtained by equating the factorised part with zero and solve it for the variable $ x $ .