Let's factor the given polynomial expression $u^3 - 7u^2 + 5u - 35$ by grouping.

Step 1: Group terms in pairs

We start by grouping the terms in pairs:

$(u^3 - 7u^2) + (5u - 35)$

Step 2: Factor out the common factor from each group

• From $u^3 - 7u^2$, we can factor out $u^2$, leaving $u - 7$.
• From $5u - 35$, we can factor out $5$, leaving $u - 7$.

Thus, we have:

$u^2(u - 7) + 5(u - 7)$

Step 3: Factor out the common binomial factor

Both terms have $(u - 7)$ as a common factor, so we can factor that out:

$(u - 7)(u^2 + 5)$

Final Factored Form:

The factored form of $u^3 - 7u^2 + 5u - 35$ is:

$(u - 7)(u^2 + 5)$

Would you like a more detailed breakdown or have any further questions about factoring? Here's a list of related questions to help you explore this topic:

1. How can you check if your factoring is correct by expanding the factors back out?
2. What is the significance of the difference of squares when factoring polynomials?
3. Can you factor polynomials of higher degrees using the same grouping method?
4. How do you handle cases where the terms do not initially have a common factor in grouping?
5. What other methods, like synthetic division, can be used for factoring cubic polynomials?

Tip: Always look for common factors in groups before attempting to factor completely. This can simplify the problem significantly.