How do you factor a binomial?
Understand the Problem
The question is asking for the method or steps to factor a binomial expression in algebra. This typically involves finding common factors or using special factoring techniques based on the form of the binomial.
Answer
Identify special pattern, factor out GCF, apply special techniques, check by expanding.
Answer for screen readers
To factor a binomial, identify any special pattern (such as a difference of squares or sum/difference of cubes), factor out the Greatest Common Factor (GCF), apply special factoring techniques if applicable, and check your work by expanding.
Steps to Solve
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Identify if the binomial fits a special pattern
Binomials often fit into common patterns like the difference of squares or the sum and difference of cubes. The common patterns to look for are:
- Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
- Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
- Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
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Factor out the Greatest Common Factor (GCF)
If the binomial does not fit a special pattern, the first step is to factor out the greatest common factor from both terms. For example, for $6x^2 - 12$, the GCF is $6$, so the factored form is $6(x^2 - 2)$.
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Apply special factoring techniques if applicable
After factoring out the GCF, see if the remaining binomial can be further factored using any of the special techniques mentioned above. For example, $x^2 - 16$ can be factored as $(x - 4)(x + 4)$ since it is a difference of squares.
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Check your work by expanding
Multiply the factors back together to ensure they equal the original binomial expression. This step helps verify that the factorization is correct.
To factor a binomial, identify any special pattern (such as a difference of squares or sum/difference of cubes), factor out the Greatest Common Factor (GCF), apply special factoring techniques if applicable, and check your work by expanding.
More Information
Factoring binomials is a common algebraic technique used in simplifying expressions and solving equations. Recognizing patterns is vital to mastering this skill.
Tips
Common mistakes include overlooking the greatest common factor (GCF) and incorrectly applying special patterns such as mixing up sum and difference of cubes.
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