How to factor 9k^2+66k+21 completely

Key takeaway: If every term shares a common factor, take it out first. It usually reveals a much friendlier trinomial hiding underneath.

Start with

$9k^2+66k+21$

Step 1: Factor out the GCF

Each term is divisible by $3$:

$3(3k^2+22k+7)$

Step 2: Factor the trinomial inside

We need two numbers that multiply to $21$ and add to $22$.

Those numbers are $21$ and $1$.

So,

$3k^2+22k+7=(3k+1)(k+7)$

Final answer

$\boxed{3(3k+1)(k+7)}$

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Pitfalls the pros know 👇 Do not stop after pulling out the $3$. The expression is not fully factored until the trinomial inside is also broken down. A quick FOIL check confirms the result.

What if the problem changes? If the middle term were $-66k$ instead of $+66k$, the inside factorization would become $(3k-1)(k-7)$, giving $3(3k-1)(k-7)$.

Tags: greatest common factor, complete factorization, trinomial factoring