Which factoring method should you use to factor the numerator of the first fraction in $\frac{x^2+6 x+8}{x^3+3 x-10} \div \frac{x^2+3 x-4}{x^3+2 x-16}$?
A. The polynomial is in simplest form.
B. Prime factorization
C. Factoring trinomials
D. Difference of squares
Answer (2)
The appropriate method to factor the numerator x 2 + 6 x + 8 is 'Factoring trinomials'. This involves finding two numbers that multiply to 8 and add to 6, allowing the trinomial to be factored as ( x + 4 ) ( x + 2 ) . Therefore, the answer is Factoring trinomials .
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The problem requires identifying the correct factoring method for the quadratic trinomial x 2 + 6 x + 8 .
Factoring trinomials involves finding two numbers that multiply to the constant term (8) and add up to the coefficient of the linear term (6).
The numbers 4 and 2 satisfy these conditions, allowing us to factor the trinomial as ( x + 4 ) ( x + 2 ) .
Therefore, the appropriate method is Factoring trinomials .
Explanation
Analyzing the Polynomial The numerator of the first fraction is the polynomial x 2 + 6 x + 8 . We need to determine the best method to factor this polynomial from the given options: Prime factorization, Factoring trinomials, and Difference of squares.
Identifying the Structure The polynomial x 2 + 6 x + 8 is a quadratic trinomial, which means it's in the form a x 2 + b x + c , where a = 1 , b = 6 , and c = 8 . The 'Factoring trinomials' method is specifically designed for polynomials of this form.
Factoring the Trinomial To factor the trinomial x 2 + 6 x + 8 , we look for two numbers that multiply to 8 and add to 6 . These numbers are 4 and 2 , since 4 × 2 = 8 and 4 + 2 = 6 . Therefore, we can factor the trinomial as ( x + 4 ) ( x + 2 ) .
Eliminating Other Methods The 'Prime factorization' method is used for factoring numbers into their prime factors, not polynomials. The 'Difference of squares' method applies to polynomials in the form a 2 − b 2 , which is not the form of our polynomial.
Conclusion Therefore, the appropriate factoring method for the numerator x 2 + 6 x + 8 is 'Factoring trinomials'.
Examples
Factoring trinomials is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to simplify complex equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to model and predict market trends, helping them make informed decisions about investments and resource allocation. Understanding factoring trinomials enables professionals to solve problems efficiently and accurately in diverse fields.
