Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ (x-1)^{2}-2(x-1) $$

Short answer

(x-1)(x-3)

Step-by-step solution

Substitute a Variable

Let’s simplify our polynomial by making a substitution. Let’s set \(u = x - 1\). This makes our polynomial \( (x-1)^2 - 2(x-1) \) become \( u^2 - 2u \).

Factor Out the Common Term

Now factor the common term \(u\) from \(u^2 - 2u\). We have: \[ u^2 - 2u = u(u - 2) \]

Substitute Back the Original Variable

Replace \(u\) back with \(x-1\). Thus, we have: \[ (x-1)((x-1) - 2) = (x-1)(x - 3) \]

Key concepts

substitution in algebra

Substitution in algebra helps simplify complex polynomials. By temporarily replacing a variable expression with another variable, the problem often becomes easier to manage. For example, in our exercise, we substituted \( u = x - 1 \). This allows us to rewrite the polynomial \( (x-1)^2 - 2(x-1) \) as \( u^2 - 2u \). Simplification through substitution isolates the core variable, making the next steps more straightforward.
Always remember to substitute back the original expression once the problem is simplified, as it helps revert the solution to the original context. This technique is valuable, especially when dealing with quadratic polynomials or radicals where substitution can convert a complex structure into a simpler equation.

common factors

Factoring out common factors is a key step in polynomial simplification. Identifying and extracting common factors from each term brings us closer to breaking down the polynomial into simpler components. In our example, the simplified polynomial \( u^2 - 2u \) has a common factor of \( u \).
This allows us to express the polynomial as: \[ u(u - 2) \].
It's easier to see common factors after substitution and simplification. Look for terms that share the same variable or number, and factor them out. This method reveals the structure of the polynomial, making further steps more obvious.

• Find common factors in each term
• Factor them out to simplify further
• Revel in how simple the polynomial becomes!

factoring techniques

Factoring techniques are crucial in solving polynomial expressions efficiently. After substitution and finding common factors, the next step is to apply these techniques properly. For instance, in the exercise, we factored \( u(u - 2) \). This process involves:
1. Identifying the simplified form.
2. Looking for patterns such as squares or simple binomials.
3. Rewriting the polynomial in its factored form.
Finally, we substitute back to get: \[ (x-1)(x - 3) \].
Using proven techniques such as substitution and factoring common factors enables us to handle complex problems step-by-step. Each step simplifies the polynomial and edges closer to a complete solution.
Remember, practice makes perfect. The more you work with these techniques, the easier they become!