Certainly! Let's factorize the expression [tex]\( p^3 - p^2 - 6p \)[/tex] step-by-step.
### Step-by-Step Solution:
1. Identify Common Factors:
The first step in factorizing any polynomial is to look for common factors in all terms of the given expression. Here, we notice that each term in [tex]\( p^3 - p^2 - 6p \)[/tex] contains the variable [tex]\( p \)[/tex].
2. Factor Out the Common Factor:
We can factor [tex]\( p \)[/tex] out from each term:
[tex]\[
p^3 - p^2 - 6p = p(p^2 - p - 6)
\][/tex]
3. Factor the Quadratic Expression:
Now, we focus on the quadratic expression [tex]\( p^2 - p - 6 \)[/tex] within the parentheses. We need to factorize [tex]\( p^2 - p - 6 \)[/tex] further.
To factorize the quadratic expression, we look for two numbers that multiply to give the constant term (-6) and add up to give the coefficient of the linear term (-1).
The quadratic expression [tex]\( p^2 - p - 6 \)[/tex] can be decomposed as follows:
[tex]\[
p^2 - p - 6 = (p - 3)(p + 2)
\][/tex]
This is because:
[tex]\[
(p - 3)(p + 2) = p^2 + 2p - 3p - 6 = p^2 - p - 6
\][/tex]
4. Combine the Factors:
Finally, substitute the factorized quadratic expression back into the original expression that we factored out in step 2:
[tex]\[
p(p^2 - p - 6) = p(p - 3)(p + 2)
\][/tex]
Hence, the fully factorized form of the expression [tex]\( p^3 - p^2 - 6p \)[/tex] is:
[tex]\[
p(p - 3)(p + 2)
\][/tex]
This is the factorized form of the polynomial given.
