Question

How do you factor \[5{x^4} - 40x + 10{x^3} - 20{x^2}\] .

Answer 1

Hint: In order to solve the question given above, you have to first use the hit and trial method to find a factor of the above equation. Then using this factor, you perform division to find a quadratic equation. In the next step you can solve the quadratic equation by either using the formula for quadratic equation or by simply splitting the middle term of the equation. You can also group the like terms in a specific manner to get the answer. This will give you your factors.

Complete step by step solution:
We have to factorize \[5{x^4} - 40x + 10{x^3} - 20{x^2}\] .
First, take \[5x\] common from the above equation, you get,
 \[5x\left( {{x^3} - 8 + 2{x^2} - 4x} \right)\] .
This can be written as: \[5x\left( {{x^3} + 2{x^2} - 4x - 8} \right)\] .
Now, we have to find one of the common factors of the above equation. For this put \[x = 2\], we get,
 \[
 {2^3} + 2{\left( 2 \right)^2} - 4\left( 2 \right) - 8 \\
 = 0 \;
 \] .
So, we get that, \[\left( {x - 2} \right)\] is one of the factors of \[{x^3} + 2{x^2} - 4x - 8\] .
Now we can find the factors of \[{x^3} + 2{x^2} - 4x - 8\] by dividing it by \[\left( {x - 2} \right)\] .
Or we can factor \[{x^3} + 2{x^2} - 4x - 8\] by grouping the terms. We will group the first and the third terms.
We get,
\[ \Rightarrow \left( {{x^3} - 4x} \right) - 2{x^2} + 8\] .
This can also be written as:
\[ \Rightarrow x\left( {{x^2} - 4} \right) - 2\left( {{x^2} - 4} \right)\]
From this we get:
\[ \Rightarrow \left( {{x^2} - 4} \right)\left( {x - 2} \right)\]
Therefore, our roots are \[5x\left( {x - 2} \right){\left( {x + 2} \right)^2}\] .
So, the correct answer is “ \[5x\left( {x - 2} \right){\left( {x + 2} \right)^2}\] ”.

Note: While solving questions similar to the one given, always remember to use hit and trial method to find one of the factors of the equation or we can simply group the like terms in the equation. This method makes solving questions much simpler and easy. Also, you can use two methods to solve a quadratic equation. 1) This method involves splitting the middle term. And 2) this method involves using the formula \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], this method is often used when the answer is in fractions or involved bigger digits.