Factoring a polynomial by grouping is a technique used when you have a polynomial with four or more terms and no single common factor for all terms. It involves grouping terms, factoring out common factors from each group, and then factoring out a common binomial.
Steps to Factor by Grouping
Here's a breakdown of the process:
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Arrange the terms: Rearrange the terms of the polynomial, if necessary, so that terms with common factors are next to each other. This might require some trial and error.
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Group the terms: Place parentheses around pairs of terms to create groups. Ensure that each group has a common factor.
- Example:
ax + ay + bx + bybecomes(ax + ay) + (bx + by)
- Example:
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Factor each group: Factor out the greatest common factor (GCF) from each group.
- Example:
(ax + ay) + (bx + by)becomesa(x + y) + b(x + y)
- Example:
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Factor out the common binomial: If you've done everything correctly, the expressions in the parentheses should now be identical. Factor out this common binomial.
- Example:
a(x + y) + b(x + y)becomes(x + y)(a + b)
- Example:
Example
Let's factor the polynomial 3x³ - 2x² - 6x + 4:
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Arrange: The terms are already arranged well.
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Group:
(3x³ - 2x²) + (-6x + 4) -
Factor each group:
x²(3x - 2) - 2(3x - 2) -
Factor out the binomial:
(3x - 2)(x² - 2)
Therefore, the factored form of 3x³ - 2x² - 6x + 4 is (3x - 2)(x² - 2).
When Grouping Doesn't Work
Sometimes, even after rearranging, grouping might not lead to a common binomial factor. In such cases, other factoring techniques (like factoring by trial and error or using special factoring formulas) might be necessary.
Key Considerations
- Rearranging: Don't be afraid to rearrange the terms to find a suitable grouping.
- Signs: Pay close attention to signs, especially when factoring out negative numbers. A negative sign outside the parentheses will change the signs inside.
- Practice: The more you practice, the better you'll become at recognizing patterns and choosing the best groupings.
