Question

How do you factor \[24{x^3} + 60{x^2} - 168x - 420\] ?

Answer 1

Hint: Here, We have the polynomial factor of the algebraic expression of the degree cube in the first term and degree square in the second term to solve using the greatest common factor and then grouping the factor.

Complete step by step solution:
The method for factoring polynomials will be factoring out the greatest common factor. When factoring in general this will also be the first thing that we should try as it will often simplify the problem. To use this method all that we do is look at all the terms and determine if there is a factor that is in common to all the terms. If there is, we will factor it out of the polynomial. Also note that in this case we are really only using the distributive law in reverse.
Given the polynomial equation, we have
 \[24{x^3} + 60{x^2} - 168x - 420\]
By greatest common factor, we get
 \[12(2{x^3} + 5{x^2} - 14x - 35)\]
Let’s find a factor, we get
 \[
  12({x^2}(2x + 5) - 7(2x + 5)) \\
  12(2x + 5)({x^2} - 7) \;
\]
Therefore, the grouping factor is \[12(2x + 5)({x^2} - 7)\] .
So, the correct answer is “ \[12(2x + 5)({x^2} - 7)\] ”.

Note: We need to remember if a polynomial has degree then it will have N roots.The algebraic expression of the degree cube is in the first term and degree square in the second term to solve using the greatest common factor. When the algebraic expression with two terms is called a binomial also known as a two term polynomial.