Choose three roots of $x^3+2 x^2-16 x-32$.
-2
1
5
To find the roots of the polynomial \(P(x) = x^3 + 2x^2 - 16x - 32\), we look for values of \(x\) such that \(P(x) = 0\). The polynomial has four terms, which suggests that we can factor it by grouping. Grouping the first two terms and the last two terms, we have:
\[P(x) = (x^3 + 2x^2) - (16x + 32)\]
Now, we factor out the common terms from each group:
\[P(x) = x^2(x + 2) - 16(x + 2)\]
Since both terms share a common factor of \((x + 2)\), we can factor that out:
\[P(x) = (x^2 - 16)(x + 2)\]
Now that we have the expression \(P(x) = (x^2 - 16)(x + 2)\), we observe that the term \((x^2 - 16)\) is a difference of squares. According to the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), where \(a = x\) and \(b = 4\). Applying this formula, we get:
\[x^2 - 16 = (x - 4)(x + 4)\]
Substituting this back into our expression for \(P(x)\), we obtain the fully factored form:
\[P(x) = (x - 4)(x + 4)(x + 2)\]
To find the roots, we set the factored polynomial equal to zero:
\[(x - 4)(x + 4)(x + 2) = 0\]
This equation holds true if any of the factors is equal to zero:
Thus, the three roots of the polynomial are \(-2\), \(-4\), and \(4\). Comparing these values to the provided list \(\{-2, 1, 5, -4, 4\}\), we confirm that the roots are \(-2\), \(-4\), and \(4\).
In structural engineering, polynomials are used to model the stress-strain behavior of materials. Finding the roots of a characteristic polynomial like the one in this problem helps engineers determine critical values, such as the points where a structure changes its behavior or reaches its load capacity. Similarly, in economics, identifying roots of a model's equations helps determine equilibrium points where market supply equals demand. Understanding how to factor and solve these equations is essential for mastering these analytical fields.