Question

6p^4+21p^3+9p^2. Factor completely

Answer 1

To factor the expression \( 6p^4 + 21p^3 + 9p^2 \), we start by looking for a common factor in all the terms.

1. **Find the Greatest Common Factor (GCF)**:
- The coefficients are 6, 21, and 9. The GCF of these numbers is 3.
- The lowest power of \( p \) present in all terms is \( p^2 \).

Therefore, the GCF of the entire expression is \( 3p^2 \).

2. **Factor out the GCF**:
\[
6p^4 + 21p^3 + 9p^2 = 3p^2(2p^2 + 7p + 3)
\]

3. **Factor the quadratic \( 2p^2 + 7p + 3 \)**:
We look for two numbers that multiply to \( 2 \cdot 3 = 6 \) (the product of the coefficient of \( p^2 \) and the constant) and add to \( 7 \) (the coefficient of \( p \)). The numbers \( 6 \) and \( 1 \) fit this requirement.

Using these numbers, we can rewrite the middle term:
\[
2p^2 + 6p + 1p + 3
\]

Then we group the terms:
\[
= (2p^2 + 6p) + (1p + 3)
\]
\[
= 2p(p + 3) + 1(p + 3)
\]

Now we can factor out the common binomial \( (p + 3) \):
\[
= (2p + 1)(p + 3)
\]

4. **Combine everything together**:
Thus, the completely factored form of the original expression is:
\[
6p^4 + 21p^3 + 9p^2 = 3p^2(2p + 1)(p + 3)
\]

So, the final answer is:
\[
\boxed{3p^2(2p + 1)(p + 3)}
\]