Problem 33 - Elementary and Intermediate Algebra

Question

Factor. $$ 3 u^{2}-18 u+27 $$

Step-by-Step Solution

Answer

The polynomial factors as \( 3(u-3)^2 \).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to find the greatest common factor (GCF) of the given polynomial terms, which are \( 3u^2 \), \(-18u\), and \(27\). The GCF of these coefficients \(3, -18,\) and \(27\) is 3.

Step 2: Factor Out the GCF

Now that we found the GCF to be 3, we factor it out of each term in the polynomial:\[3u^2 - 18u + 27 = 3(u^2 - 6u + 9)\]

Step 3: Recognize the Perfect Square Trinomial

Inside the parentheses, we have the expression \( u^2 - 6u + 9 \). This can be identified as a perfect square trinomial. It matches the form \( a^2 - 2ab + b^2 = (a-b)^2 \), where \( u^2 = a^2 \), \( -6u = -2ab \), and \( 9 = b^2 \).

Step 4: Factor the Perfect Square Trinomial

Since \( a^2 - 2ab + b^2 = (a-b)^2 \), and here \( b = 3 \) (since \( b^2 = 9 \) implies \( b = 3 \)), the trinomial can be written as:\[u^2 - 6u + 9 = (u-3)^2\]

Step 5: Write the Final Factored Form

Substitute the factored form from Step 4 back into the expression from Step 2:\[3(u^2 - 6u + 9) = 3(u-3)^2\]Thus, the completely factored form of the original polynomial is \( 3(u-3)^2 \).

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest number that divides each term in a given expression without leaving a remainder. Specifically for polynomials, finding the GCF is crucial as it simplifies the initial steps of factoring. Let's look at what this means in the context of our polynomial.
Consider the polynomial terms in the expression: \(3u^2\), \(-18u\), and \(27\). The coefficients are 3, -18, and 27. The GCF of these numbers is determined by identifying the highest number that can divide each of these coefficients:

• 3 divides 3.
• 3 divides -18 (since \(-18 \div 3 = -6\)).
• 3 divides 27 (since \(27 \div 3 = 9\)).

So, the GCF of the polynomial is 3. This means you can factor a 3 out of the entire polynomial, simplifying subsequent steps. Understanding and finding the GCF is the first crucial step in breaking down polynomial expressions effectively.

Perfect Square Trinomial

Recognizing a perfect square trinomial can make factoring simpler and more straightforward. A perfect square trinomial takes the form \(a^2 - 2ab + b^2 = (a-b)^2\). Let's see how we apply this to our polynomial.
After factoring out the GCF, we are left with \(u^2 - 6u + 9\). To recognize it as a perfect square trinomial, we check:

• \(u^2\) is \(a^2\), so \(a = u\).
• -6u matches \(-2ab\) where \(b = 3\), satisfying \(-6u = -2 \cdot u \cdot 3\).
• 9 is \(b^2\), confirming that \(b = 3\).

This trinomial can be rewritten as \((u-3)^2\). Thus, rather than dealing with separate terms, you can condense it into a squared binomial, which simplifies further manipulations or calculations.

Polynomial Factoring Steps

Factoring polynomials follows a set of systematic steps to simplify expressions. Here’s how you can approach factoring a polynomial like \(3u^2 - 18u + 27\):
**Step 1: Identify the GCF**Find the greatest common factor of all terms. In this case, it's 3, which you'll factor out of the polynomial.
**Step 2: Factor out the GCF**Rewriting the polynomial by factoring out the GCF: \(3(u^2 - 6u + 9)\).
**Step 3: Recognize Perfect Square Trinomials**Look inside the parentheses: \(u^2 - 6u + 9\) is a perfect square trinomial, which simplifies to \((u-3)^2\).
**Step 4: Final Factored Form**Insert the simplified expression back into the original equation after factoring out the GCF:\[3(u^2 - 6u + 9) = 3(u-3)^2\]Thus leading to the final factored form of \(3(u-3)^2\).
These steps provide a reliable framework for tackling polynomial expressions, ensuring efficient and accurate simplification.