How are Polynomials factored by grouping? Explain and give example.
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Factoring polynomials by grouping is a method used to factor expressions with four or more terms. The main idea is to group the terms into pairs and then factor out a common factor from each pair. This creates two new binomial expressions, which can then be further factored if possible.
Let's go through an example to understand the process better. Consider the polynomial expression:
2x^3 + 4x^2 - 3x - 6.
Step 1: Group the terms in pairs. In this case, we can pair the first two terms and the last two terms:
(2x^3 + 4x^2) - (3x + 6).
Step 2: Factor out the greatest common factor (GCF) from each group. Take out 2x^2 from the first pair and -3 from the second pair:
2x^2(x + 2) - 3(x + 2).
Step 3: Observe that we now have a common binomial factor, (x + 2), in both terms.
(2x^2 - 3)(x + 2).
The factored form of the original polynomial is (2x^2 - 3)(x + 2).
To check if the factoring is correct, you can expand the factored expression using the distributive property:
(2x^2 - 3)(x + 2) = 2x^3 + 4x^2 - 3x - 6.
This confirms that we have factored the polynomial correctly.
In summary, to factor a polynomial by grouping, you group the terms, factor out the GCF from each group, and then look for common binomial factors among the terms. By following these steps, you can factor polynomials efficiently.