One of the factors of the polynomial $x^3+5 x^2+6 x$ is $\left(x^2+3 x\right)$. What is the other factor?
The objective is to find the missing factor \(R(x)\) such that \(P(x) = x^3 + 5x^2 + 6x\) is expressed as the product of the known factor \(Q(x) = x^2 + 3x\) and the unknown factor \(R(x)\). We are given that \(P(x) = Q(x) \cdot R(x)\), which means \(x^3 + 5x^2 + 6x = (x^2 + 3x) \cdot R(x)\). Since \(P(x)\) is a cubic polynomial and \(Q(x)\) is a quadratic one, \(R(x)\) must be a linear polynomial of the form \(ax + b\).
To find \(R(x)\), we can factor out common terms from the polynomial \(P(x)\). First, observe that each term in \(P(x)\) has an \(x\):
\[P(x) = x(x^2 + 5x + 6)\]
Next, factor the quadratic expression \(x^2 + 5x + 6\). We look for two numbers that multiply to 6 and add to 5. These numbers are 2 and 3. Thus,
\[x^2 + 5x + 6 = (x+2)(x+3)\]
Substituting this back into the expression for \(P(x)\), we get:
\[P(x) = x(x+2)(x+3)\]
Now, let's look at the given factor \(Q(x) = x^2 + 3x\). We can factor this by taking out the common \(x\):
\[Q(x) = x(x+3)\]
We know that \(P(x) = Q(x) \cdot R(x)\), so substituting the factored forms:
\[x(x+2)(x+3) = [x(x+3)] \cdot R(x)\]
To isolate \(R(x)\), divide both sides by \(x(x+3)\):
\[R(x) = \frac{x(x+2)(x+3)}{x(x+3)} = x+2\]
Thus, the other factor is \(x+2\). Comparing this result with the given options, we identify the correct choice.
Polynomial factoring is a fundamental skill in algebra with many real-world applications. For instance, in engineering, when designing structures or mechanical systems, engineers often use polynomials to model the stress or load distribution on a beam. By factoring these polynomials, they can identify critical points or specific components that contribute to the overall strength or failure of the system. Just as we factored \(x^3 + 5x^2 + 6x\) to isolate a specific structural factor \(x+2\), engineers factor complex equations to simplify their analysis and make better design decisions.