Factor special products
Understand the Problem
The question is asking about how to factor special products, which typically includes identities like the difference of squares, perfect square trinomials, and the sum/difference of cubes. To solve it, we will identify the type of special product and apply the corresponding factoring technique.
Answer
The answer depends on the specific expression being factored; for $x^2 - 9$, it would be $(x - 3)(x + 3)$.
Answer for screen readers
The answer will depend on the specific expression you are factoring. For example, if factoring $x^2 - 9$, the answer is $(x - 3)(x + 3)$.
Steps to Solve
- Identify the special product type
First, determine which special product identity applies to the expression you need to factor. Common types are:
- Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
- Perfect square trinomial: $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$
- Sum or difference of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ or $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
- Apply the appropriate factoring technique
Once you identify the type, apply the corresponding factoring technique. For example:
- For the difference of squares, if your expression is $x^2 - 9$, recognize it as $x^2 - 3^2$ and factor it as $(x - 3)(x + 3)$.
- For a perfect square trinomial like $x^2 + 6x + 9$, recognize it as $(x + 3)^2$.
- Verify your factored expression
After factoring, always double-check by expanding your factored expression to ensure it matches the original expression. For example, if you factor $x^2 - 9$ as $(x - 3)(x + 3)$, expand it:
$$(x - 3)(x + 3) = x^2 - 9$$
This confirms your factorization is correct.
The answer will depend on the specific expression you are factoring. For example, if factoring $x^2 - 9$, the answer is $(x - 3)(x + 3)$.
More Information
Factoring special products is a crucial skill in algebra, as it simplifies expressions and solves equations efficiently. Each special product identity helps in understanding how polynomials can be manipulated and recognized.
Tips
- Failing to recognize the best identity for factoring. Always examine the structure of the expression thoroughly before applying an identity.
- Not checking your work by expanding the factored form back into the original expression.
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