Factoring the polynomial x³ + 11x² + 3x + 33 by grouping is a systematic approach that can simplify the process of determining its factors. Here’s a step-by-step guide on how to do it:
- Group the Terms: First, we will group the polynomial into two pairs. This can help us factor them separately. So, we can group it like this:
- Factor Out Common Elements: Next, we will factor out the common elements from each grouped pair:
- From the first group (x³ + 11x²), we can factor out x²:
x²(x + 11) - From the second group (3x + 33), we can factor out 3:
3(x + 11) - Rewrite the Expression: Now, we can rewrite our expression after factoring:
- Factor Out the Common Binomial: Notice that (x + 11) is now a common factor in both terms, so we factor that out:
(x³ + 11x²) + (3x + 33)x²(x + 11) + 3(x + 11)(x + 11)(x² + 3)Thus, the factors of the given polynomial x³ + 11x² + 3x + 33 by grouping are:
(x + 11)(x² + 3)
These factors represent the polynomial in a more manageable form. It’s essential to further analyze (x² + 3) since it does not factor nicely over the reals. Therefore, the final factored form we can use for most applications is:
(x + 11)(x² + 3)This method not only simplifies the polynomial but also enhances our understanding of its structure.