Question

How do you factor completely: \[ - 5{x^3} + 10{x^2}\] ?

Answer 1

Hint: An expression is completely factored when no further factoring is possible. The possibility of factoring by grouping exists when an expression contains four or more terms. Factoring polynomials involves breaking up a polynomial into simpler terms (the factors) such that when the terms are multiplied together, they equal the original polynomial. Here, in the given equation, we need to group the common factors and factor out the given terms.

Complete step by step solution:
Completely factor means to continuously factor terms until they are in simple terms, meaning you are no longer able to factor. Factoring is a process that changes a sum or difference of terms to a product of factors. A prime expression cannot be factored. The greatest common factor is the greatest factor common to all terms. To find the greatest common factor (GCF) between monomials, take each monomial and write its prime factorization. An expression is completely factored when no further factoring is possible.
Let us write the given equation:
 \[ - 5{x^3} + 10{x^2}\]
To factor the given equation. We need to factor out -1 as:
 \[ \Rightarrow - \left( {5{x^3} - 10{x^2}} \right)\]
Now, factor the common terms, we get:
 \[ \Rightarrow - 5{x^2}\left( {x - 2} \right)\]
Thus,
 \[ - 5{x^3} + 10{x^2} = - 5{x^2}\left( {x - 2} \right)\]
So, the correct answer is “\[5{x^2}\left( {x - 2} \right)\]”.

Note: The key point to find the equations using factoring method i.e., if the given equation is of the form \[a{x^2} + bx + c\] , then we need to find two integers whose product is equal to c and the sum is equal to b using AC method. Then solve each factor obtained by setting it to zero by this we can get the value of b of both the factors.