- Analyze the polynomial \(3x^3 + 6x^2 + 5x + 10\), which contains four terms.
- Compare the characteristics of the given options: perfect-square trinomial (3 terms), difference of squares (2 terms), factor by grouping (often 4 terms), and sum of cubes (2 terms).
- Apply factor by grouping to verify it works by grouping terms into \((3x^3 + 6x^2)\) and \((5x + 10)\).
- Conclude that the correct method is \(\boxed{\text{factor by grouping}}\).
Explanation
- Analyzing the Polynomial Structure
The polynomial given is \(3x^3 + 5x + 6x^2 + 10\). To identify the appropriate factoring method, we first count the number of terms. The expression has four distinct terms: \(3x^3\), \(5x\), \(6x^2\), and \(10\). Analyzing the options, a perfect-square trinomial generally consists of three terms, a difference of squares consists of two terms, and a sum of cubes consists of two terms. Factor by grouping, however, is a technique specifically designed for polynomials with an even number of terms, most commonly four.
- Testing and Confirming the Factoring Method
We can test the "factor by grouping" method by rearranging the terms to group them in pairs. Let's reorder the expression to group terms with common factors:
\[3x^3 + 6x^2 + 5x + 10\]
Now, group the first two terms and the last two terms:
\[(3x^3 + 6x^2) + (5x + 10)\]
Extract the greatest common factor from each group:
- From the first group \((3x^3 + 6x^2)\), we factor out \(3x^2\) to get \(3x^2(x + 2)\).
- From the second group \((5x + 10)\), we factor out \(5\) to get \(5(x + 2)\).
The expression becomes:
\[3x^2(x + 2) + 5(x + 2)\]
Since both terms share a common factor of \((x + 2)\), we can factor it out:
\[(3x^2 + 5)(x + 2)\]
Because the grouping method effectively factors the polynomial, it is the correct approach.
- Conclusion
Based on the analysis, the factoring method suitable for the given four-term polynomial is factor by grouping. Other methods like the perfect-square trinomial, difference of squares, and sum of cubes do not align with the four-term structure. Thus, the correct choice is factor by grouping.
Examples
Factoring by grouping is essential in algebra when solving equations of higher degrees, such as cubic equations, which are often used in physics to model volume or density calculations. For example, if the volume of a box is modeled by a polynomial like \(V = 3x^3 + 6x^2 + 5x + 10\), factoring it allows you to find the dimensions of the box in terms of \(x\), which is crucial for engineering and design tasks. Mastering this technique helps simplify complex expressions into manageable parts, facilitating easier solutions for real-world problems.