Sure, let's factorize the expression [tex]\( p^3 - p^2 - 6p \)[/tex] step-by-step.
1. Identify a common factor: The first step in factorizing the expression is to identify any common factors in all terms. Here, each term contains [tex]\( p \)[/tex], so we can factor [tex]\( p \)[/tex] out of the entire expression:
[tex]\[
p^3 - p^2 - 6p = p(p^2 - p - 6)
\][/tex]
2. Factor the quadratic expression: Now, we need to factorize the quadratic expression [tex]\( p^2 - p - 6 \)[/tex]. We look for two numbers that multiply to give the constant term (-6) and add to give the coefficient of the linear term (-1).
The pair of numbers that fit these conditions are -3 and 2, since:
[tex]\[
(-3) \cdot (2) = -6 \quad \text{and} \quad (-3) + 2 = -1
\][/tex]
3. Rewrite the middle term: Using these numbers, we can rewrite the quadratic expression [tex]\( p^2 - p - 6 \)[/tex] as:
[tex]\[
p^2 - p - 6 = p^2 - 3p + 2p - 6
\][/tex]
4. Group and factor by grouping: Next, group the terms to factor by grouping:
[tex]\[
p^2 - 3p + 2p - 6 = (p^2 - 3p) + (2p - 6)
\][/tex]
Factor out the common terms in each group:
[tex]\[
(p(p - 3)) + (2(p - 3))
\][/tex]
Now, factor out the common binomial factor [tex]\( p - 3 \)[/tex]:
[tex]\[
(p - 3)(p + 2)
\][/tex]
5. Combine all factors: Putting it all together, we have:
[tex]\[
p(p^2 - p - 6) = p(p - 3)(p + 2)
\][/tex]
Therefore, the factorization of the expression [tex]\( p^3 - p^2 - 6p \)[/tex] is:
[tex]\[
p(p - 3)(p + 2)
\][/tex]
