To factor out the greatest common factor (GCF) from the expression:

\[
30k^4 - 15k^3 + 70k^2 - 35k
\]

### Step 1: Identify the GCF
First, identify the GCF of the coefficients (30, 15, 70, 35) and the powers of \(k\) in each term:

- The GCF of the coefficients \(30, 15, 70,\) and \(35\) is 5.
- The smallest power of \(k\) present in all terms is \(k\).

So, the GCF is \(5k\).

### Step 2: Factor out the GCF
Now, factor \(5k\) out of each term:

\[
30k^4 - 15k^3 + 70k^2 - 35k = 5k(6k^3 - 3k^2 + 14k - 7)
\]

Thus, the factored form of the expression is:

\[
5k(6k^3 - 3k^2 + 14k - 7)
\]

Would you like any further details or have any questions?

Here are some related questions you might find helpful:
1. Can the expression inside the parentheses be factored further?
2. What happens if we expand the factored expression to check our work?
3. How does factoring help in solving polynomial equations?
4. What would the expression look like if we factored out a different common factor?
5. Can we use the factored expression to find roots more easily?

**Tip:** Always double-check your factored expression by distributing the GCF back through the terms to ensure it matches the original expression.

Learn how to factor out the greatest common factor (GCF) from the polynomial expression 30k^4 - 15k^3 + 70k^2 - 35k. This problem involves identifying the GCF of coefficients and powers of k, and factoring it out step-by-step. Suitable for students in Grades 9-12 studying polynomial factorization.

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