Question

How do you factor \[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)\]?

Answer 1

Hint: This type of problem is based on the concept of factoring a polynomial. First, we have to consider the whole polynomial with degree 3. We have looked for any common terms which is \[-6{{x}^{2}}\]. Take \[-6{{x}^{2}}\]common outside the bracket. We get 1-x to be one factor of the given polynomial. Now, we can write \[6{{x}^{2}}\]as 6x and x. Thus, we get three factors of the given polynomial.

Complete step by step solution:
According to the question, we are asked to find the factors of \[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)\].
We have been given the polynomial is \[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)\]. -----(1)
The given polynomial is of degree 3 and variable x.
We can express the polynomial (1) as
\[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)=-6{{x}^{2}}-\left( -6{{x}^{2}}\times x \right)\]
Here, we find that \[-6{{x}^{2}}\] is common in both the terms of the polynomial.
Let us take \[-6{{x}^{2}}\] common from both the terms.
\[\Rightarrow -6{{x}^{2}}-\left( -6{{x}^{3}} \right)=-6{{x}^{2}}\left( 1-x \right)\]
But we know that a factor should be of degree 1.
Thus, we have to split \[-6{{x}^{2}}\] into two terms.
We know that \[{{x}^{2}}=x\times x\]. Therefore, we can write the polynomial as
\[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)=-6x\times x\left( 1-x \right)\]
Here, we find that the polynomial is converted as a product of three linear polynomials.
These are the factors of the given polynomial.
Therefore, the factors of \[-6{{x}^{2}}-\left( -6{{x}^{3}} \right)\] are -6x, x and 1-x.

Note: We can also solve this question by taking \[6{{x}^{2}}\] common from the given polynomial and solve the rest using the same method mentioned above. We cannot consider \[-6{{x}^{2}}\] to be a factor of the given polynomial since it is of degree 2. Avoid calculation mistakes based on sign convention. Since the polynomial is of degree 3, we get 3 factors and not less than 3.