How do you factor an expression completely?

Understand the Problem

The question is asking for the process of factoring an expression completely, which involves rewriting it as a product of its factors. We will typically look for common factors, apply techniques such as grouping, and use factoring formulas where applicable.

Answer

The completely factored form is $ (x + 2)(x + 3) $.

Steps to Solve

Identify the expression to factor
Start with the given expression. For example, consider the expression ( x^2 + 5x + 6 ).
Look for common factors
Check if there are any common factors across all terms. In the example ( x^2 + 5x + 6 ), there are no common factors.
Apply the factoring method
Since there are no common factors, we can try to factor the expression by finding two numbers that multiply to give the constant term (6) and add to give the linear coefficient (5). The numbers 2 and 3 satisfy this.
Write the factored form
Using the numbers from the previous step, we can express the original expression as a product:
$$ (x + 2)(x + 3) $$
Verify the factorization
To ensure the factorization is correct, multiply the factors back together:
$$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$$
This confirms that the factorization is correct.

The completely factored form of the expression ( x^2 + 5x + 6 ) is ( (x + 2)(x + 3) ).

More Information

Factoring is an important algebraic skill that helps simplify expressions and solve equations. This particular expression is a quadratic, which can often be factored when it has rational roots.

Tips

Missing factors: Sometimes, students overlook that not all expressions can be factored completely, such as prime polynomials. Always double-check if a factorization is possible.
Incorrect pairs of numbers: Selecting the wrong pair of numbers that do not satisfy the multiplication and addition conditions can lead to incorrect factorizations.

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