How do you factor the expression 7x^3 + 21x^2 + 3x + 9 by grouping, and what does the resulting expression look like?

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Factoring by Grouping the Expression 7x³ + 21x² + 3x + 9

To factor the expression 7x³ + 21x² + 3x + 9 by grouping, we will follow these steps:

Step 1: Group the Terms

We start by grouping the first two terms and the last two terms:

(7x³ + 21x²) + (3x + 9)

Step 2: Factor Out the Common Factors

Next, we factor out the common factors from each group:

  • From the first group (7x³ + 21x²), the common factor is 7x²:
    7x²(x + 3)
  • From the second group (3x + 9), the common factor is 3:
    3(x + 3)

Step 3: Combine the Factored Groups

Now we can rewrite the expression as:

7x²(x + 3) + 3(x + 3)

Step 4: Factor Out the Common Binomial Factor

Notice that (x + 3) is a common factor in both terms:

(x + 3)(7x² + 3)

Conclusion

The resulting factored expression of 7x³ + 21x² + 3x + 9 by grouping is:

(x + 3)(7x² + 3)

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