- Identify the potential factors by testing roots \(-2\) and \(-4\) in the polynomial.
- Expand the expression \((x+2)(x+4)(x^2+25)\) to verify it matches the original polynomial \(x^4 + 6x^3 + 33x^2 + 150x + 200\).
- Confirm that the expanded result is identical to the given polynomial.
- The correct factored form is \(\boxed{(x+2)(x+4)(x^2+25)}\).
Explanation
- Analyzing the polynomial and testing factors
The objective is to find the factored form of the polynomial \(p(x) = x^4 + 6x^3 + 33x^2 + 150x + 200\). We are given four options to choose from. To solve this, we can either test the roots of the potential factors or expand the given options to see which matches the original polynomial. We first test the roots of the factors \((x+2)\) and \((x+4)\) by checking if \(-2\) and \(-4\) are roots of the polynomial. Evaluating \(p(-2)\), we get \((-2)^4 + 6(-2)^3 + 33(-2)^2 + 150(-2) + 200 = 16 - 48 + 132 - 300 + 200 = 0\), which confirms that \((x+2)\) is a factor. Evaluating \(p(-4)\), we get \((-4)^4 + 6(-4)^3 + 33(-4)^2 + 150(-4) + 200 = 256 - 384 + 528 - 600 + 200 = 0\), which confirms that \((x+4)\) is a factor.
- Expanding and verifying the factors
Since \((x+2)\) and \((x+4)\) are factors, the product of these two is \((x+2)(x+4) = x^2+6x+8\). Dividing the polynomial \(p(x)\) by \((x^2+6x+8)\) using polynomial division or by comparing it with the remaining quadratic factors in the choices, we can determine the correct answer. The remaining quadratic factor in the first option is \((x^2+25)\). Expanding \((x^2+6x+8)(x^2+25)\), we get \(x^2(x^2+25) + 6x(x^2+25) + 8(x^2+25) = x^4 + 25x^2 + 6x^3 + 150x + 8x^2 + 200\). Simplifying this expression results in \(x^4 + 6x^3 + (25+8)x^2 + 150x + 200 = x^4 + 6x^3 + 33x^2 + 150x + 200\). This perfectly matches the original polynomial.
- Conclusion
Based on our verification, the correct factored form of the polynomial \(x^4 + 6x^3 + 33x^2 + 150x + 200\) is \((x+2)(x+4)(x^2+25)\). Thus, the correct option is the first one. The other options are incorrect as they do not expand to the original polynomial. Therefore, the factored form is \((x+2)(x+4)(x^2+25)\).
Examples
Polynomial factoring is a fundamental skill in algebra with practical applications in fields such as engineering, physics, and computer science. For example, when analyzing the motion of an object, one might end up with a polynomial equation where finding the roots (factoring the expression) helps determine the specific times when an object hits the ground or reaches a certain height. Mastering this technique allows students to simplify complex expressions and solve equations that describe real-world phenomena.