Question

How do you factor \[{x^3} - 12{x^2} + 35x\] ?

Answer 1

Hint: To solve this question, we first need to split \[12{x^2}\] into two parts, so that pairing can be done for two terms. Next, pair up the remaining terms together. Further, find the common terms, and simplify the expression until there are only two terms remaining.

Complete step-by-step answer:
The Given Polynomial is a Cubic Trinomial. Our objective is to factorize this polynomial into simpler terms.
In this polynomial, the first thing we need to do is split the central term, i.e., \[12{x^2}\] into two terms, so that factorization can be performed. While splitting, we need to keep in mind that the sum of the coefficients of these terms equals to the coefficient of this term, i.e., 12, and their product equals to the coefficient of the term with the lowest degree, if the coefficient of that term is greater than the coefficient of the original term, i.e., 35.
Thus, we can split \[12{x^2}\] into \[7{x^2}\] and \[5{x^2}\] .
We can obtain the two pairs as:
  \[({x^3} - 7{x^2}) - (5{x^2} - 35x)\]
Now for each bracket, we need to take the common factor out of the bracket. For e.g., in \[({x^3} - 7{x^2})\] , the common factor that divides both terms inside the brackets is \[{x^2}\] .
After taking the common factors out of the bracket, we get:
  \[{x^2}(x - 7) - 5x(x - 7)\] .
Now, we also observe that in both the larger terms, i.e., \[{x^2}(x - 7)\] and \[5x(x - 7)\] , we have a common factor, which is \[(x - 7)\] . Thus, \[(x - 7)\] can be taken as common, and we can apply the distributive property to simplify this polynomial further.
On applying distributive property, i.e., \[ab + ac = a(b + c)\] , we get:
  \[{x^2}(x - 7) - 5x(x - 7) = ({x^2} - 5)(x - 7)\]
Thus, the final factorization of the polynomial \[{x^3} - 12{x^2} + 35x\] is \[({x^2} - 5)(x - 7)\] .
So, the correct answer is “ \[({x^2} - 5)(x - 7)\] ”.

Note: In this question, terms need to be paired correctly. It needs to be remembered that to factorize cubic polynomials, the middle term should be split so that the sum of coefficients of the resulting terms equals to the coefficient of the original term, and the product of those coefficients is equal to the coefficient of the term with a degree immediately lower to the original term, if the coefficient of that term is greater than the coefficient of the original term. Any other combination of pairing will result in an incorrect answer.