Factor each polynomial. Check your work by distributing. If a polynomial cannot be factored, write "prime" 6a^2 + 27

Let's begin by factoring the polynomial:

Given expression: 

\[
6a^2 + 27
\]

### Step 1: Find the greatest common factor (GCF).
The GCF of \(6a^2\) and \(27\) is \(3\), since 3 is the largest number that divides both 6 and 27 evenly.

### Step 2: Factor out the GCF.

\[
6a^2 + 27 = 3(2a^2 + 9)
\]

### Step 3: Check if \(2a^2 + 9\) can be factored further.
- \(2a^2 + 9\) cannot be factored any further because it does not fit any common factorization patterns (such as the difference of squares, perfect square trinomials, or sum/difference of cubes).

Thus, the fully factored form of the polynomial is:

\[
3(2a^2 + 9)
\]

### Step 4: Check your work by distributing.
Distribute the \(3\) back into the expression:

\[
3(2a^2 + 9) = 3 \cdot 2a^2 + 3 \cdot 9 = 6a^2 + 27
\]

Since we arrived back at the original expression, the factoring is correct.

### Final Answer:
The factored form is \(3(2a^2 + 9)\).

Would you like any further details or have any questions?

Here are five related questions to expand on this:
1. How do you determine the greatest common factor of two terms?
2. What are the common methods for factoring polynomials?
3. Can every polynomial be factored into simpler polynomials?
4. What is the difference between factoring and expanding an expression?
5. How can you recognize special factoring patterns, such as perfect squares or differences of squares?

**Tip:** Always double-check your factoring by distributing back to the original expression.

Factor each polynomial. Check your work by distributing. If a polynomial cannot be factored, write 'prime' 6a^2 + 27.

Learn how to factor the polynomial 6a^2 + 27 by finding the greatest common factor (GCF) and using the distributive property to check the work. This step-by-step guide is suitable for students in Grades 8-10 and covers core algebraic concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation