Factor the polynomial x² + 2x - 3x - 6 by double grouping. 1. Group terms with common factors. 2. Factor the GCF from each group. 3. Write the polynomial as a product of binomials. Factor the polynomial x² + 2x - 3x - 6 by double grouping. 1. Group terms with common factors. 2. Factor the GCF from each group. 3. Write the polynomial as a product of binomials.
Understand the Problem
The question involves factoring a polynomial using the double grouping method, which includes grouping terms, factoring out the greatest common factor (GCF), and expressing the polynomial as a product of binomials.
Answer
The factored form of the polynomial is $$(x + 2)(x - 3)$$.
Answer for screen readers
The factored form of the polynomial is $$(x + 2)(x - 3)$$.
Steps to Solve
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Group the terms Group the polynomial into two parts with common factors: $$(x^2 + 2x) + (-3x - 6)$$
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Factor out the GCF from each group Identify the greatest common factor (GCF) for both groups:
- For the first group $(x^2 + 2x)$, the GCF is $x$: $$x(x + 2)$$
- For the second group $(-3x - 6)$, the GCF is $-3$: $$-3(x + 2)$$
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Combine the factored groups Now, rewrite the expression as: $$x(x + 2) - 3(x + 2)$$ Notice that $(x + 2)$ is a common factor.
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Factor out the common binomial Combine the terms: $$(x + 2)(x - 3)$$
The factored form of the polynomial is $$(x + 2)(x - 3)$$.
More Information
Factoring by grouping is a useful technique for polynomials to simplify expressions and solve equations. This method works especially well when a polynomial has four terms.
Tips
- Forgetting to include the sign: It's important to carry the negative sign when factoring out the GCF from terms like $-3x - 6$.
- Neglecting to check for common factors in both groups: Always ensure that both groups share the same binomial when factoring this way.
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