Factoring polynomials is a fundamental skill in algebra that involves expressing a polynomial as a product of its irreducible factors. Factoring completely means breaking down a polynomial into its simplest form, where it cannot be further factored.

In this article, we will guide you through the steps to factor completely the expression 9x2 42x 49. We will provide a clear explanation of the techniques used and illustrate each step with detailed examples.

By following the steps outlined in this comprehensive guide, you will gain a thorough understanding of how to factor completely expressions of this type and enhance your overall algebra skills.

Identifying Common Factors

Factoring Out the Greatest Common Factor (GCF)

The first step in factoring completely is to identify and factor out the greatest common factor (GCF) of all the terms in the expression. The GCF is the largest factor that divides evenly into each term.

In this case, the GCF of 9x2, 42x, and 49 is 1, as there is no common factor other than 1 that divides into all three terms. Therefore, we cannot factor out any common factors at this stage.

Factoring by Grouping

Since there are no common factors to factor out, we will use the factoring by grouping technique. This method involves grouping terms that have common factors and factoring them out.

In this case, we can group the first two terms, 9x2 and 42x, since they both have a common factor of 9x. We can then factor out 9x from each term to get:

“`
9x2 + 42x = 9x(x + 4)
“`

Factoring the Remaining Polynomial

The remaining polynomial, x + 4, is a linear expression and cannot be factored further. Therefore, the completely factored expression is:

“`
9x2 + 42x = 9x(x + 4)
“`

Conclusion

In summary, factoring completely 9x2 42x 49 involves identifying the greatest common factor, which is 1 in this case, and then using the factoring by grouping technique to factor out the common factor of 9x. The remaining polynomial, x + 4, is a linear expression and cannot be factored further. Therefore, the completely factored expression is 9x(x + 4).

By understanding and applying these factoring techniques, you can effectively factor completely polynomial expressions and improve your overall mathematical abilities.

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