- Group the four terms into two pairs: \((x^3 + 4x^2)\) and \((-9x - 36)\).
- Extract the common factor from each pair: \(x^2(x + 4) - 9(x + 4)\).
- Recognize the common binomial factor \((x + 4)\) and factor it out to get \((x^2 - 9)(x + 4)\).
- Compare the result with the given choices, which corresponds to option \(\boxed{\text{a}}\).
Explanation
- Analysis of the Polynomial and Method
The objective is to factor the polynomial \(x^3 + 4x^2 - 9x - 36\) using the method of factoring by grouping. Factoring by grouping is a technique used for polynomials with four terms, where we pair the terms to identify common factors. The given polynomial is \(P(x) = x^3 + 4x^2 - 9x - 36\).
- Grouping and Factoring Terms
To begin the factoring process, we group the four terms of the polynomial into two pairs: \((x^3 + 4x^2)\) and \((-9x - 36)\). Now, we extract the greatest common factor (GCF) from each pair.
For the first pair \((x^3 + 4x^2)\), the GCF is \(x^2\), so we can write:
\[x^3 + 4x^2 = x^2(x + 4)\]
For the second pair \((-9x - 36)\), the GCF is \(-9\), so we can write:
\[-9x - 36 = -9(x + 4)\]
Combining these, the entire expression becomes:
\[x^2(x + 4) - 9(x + 4)\]
- Identifying the Final Factored Form
Now that we have the expression \(x^2(x + 4) - 9(x + 4)\), we observe a common binomial factor, which is \((x + 4)\). We factor out \((x + 4)\) from both parts of the expression:
\[x^2(x + 4) - 9(x + 4) = (x^2 - 9)(x + 4)\]
We have now factored the expression into \((x^2 - 9)(x + 4)\). We compare this result with the provided options:
A\((x^2 - 9)(x + 4)\)
B\((x^2 + 9)(x + 4)\)
C\((x^2 + 9)(x - 4)\)
D\((x^2 - 9)(x - 4)\)
The resulting expression matches option 'a'.
- Conclusion and Final Answer
The factorization of the polynomial \(x^3 + 4x^2 - 9x - 36\) using the grouping method results in \((x^2 - 9)(x + 4)\). This corresponds to option a. Thus, the final answer is \(\boxed{\text{a}}\).
Examples
Factoring polynomials is a fundamental skill in algebra with many real-world applications. For example, in engineering and physics, factoring can simplify complex expressions describing the trajectory of a projectile or the distribution of forces in a bridge structure. By simplifying these equations, engineers can more easily solve for critical values, such as the time when an object hits the ground or the points where stress is greatest. Mastering techniques like grouping ensures you can efficiently simplify and interpret these mathematical models.