- Test possible rational roots using the Rational Root Theorem and find that \(-2\) is a root.
- Divide the polynomial \(h(x) = x^3 - 4x^2 - 3x + 18\) by \((x+2)\) to obtain the factored form \(h(x) = (x+2)(x-3)^2\).
- Identify the roots from the factored form as \(-2, 3,\) and \(3\).
- Conclude that there are 3 real number solutions, which matches option (b).
- The final answer is \(\boxed{b}\).
Explanation
- Analysis of the Polynomial Roots
The given function is a cubic polynomial:
\[h(x) = x^3 - 4x^2 - 3x + 18.\]
According to the Fundamental Theorem of Algebra, a cubic polynomial has exactly 3 roots in the complex plane. Our objective is to determine how many of these roots are real. To do this, we can look for rational roots using the Rational Root Theorem. Possible rational roots are factors of the constant term (18) divided by factors of the leading coefficient (1). Thus, the candidates are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18\).
- Factoring the Polynomial
Let us test the candidates to see if any are roots of the function.
Testing \(x = -2\):
\[h(-2) = (-2)^3 - 4(-2)^2 - 3(-2) + 18 = -8 - 16 + 6 + 18 = 0.\]
Since \(h(-2) = 0\), we know that \((x + 2)\) is a factor of the polynomial. We can perform polynomial division or synthetic division to divide \(h(x)\) by \((x + 2)\) to find the remaining quadratic factor:
\[x^3 - 4x^2 - 3x + 18 = (x+2)(x^2 - 6x + 9).\]
The quadratic factor \(x^2 - 6x + 9\) can be factored further as:
\[x^2 - 6x + 9 = (x-3)(x-3) = (x-3)^2.\]
Therefore, the function can be fully factored as \(h(x) = (x+2)(x-3)^2\).
- Identifying the Roots
From the factored form \(h(x) = (x+2)(x-3)^2\), we can identify the roots of the equation \(h(x) = 0\). The roots are:
\(x_1 = -2\)
\(x_2 = 3\)
\(x_3 = 3\)
Since \(x_1\), \(x_2\), and \(x_3\) are all real numbers, the polynomial has 3 real roots (where 3 is a root with a multiplicity of 2). Therefore, the function has 3 real number solutions. This corresponds to option (b).
Examples
Polynomial functions appear frequently in engineering and physics, for example, to model the trajectory of a projectile or the deformation of a material under load. In these cases, finding the real roots of a polynomial helps determine where an object hits the ground or where the stress-strain curve crosses critical thresholds. Understanding the number of real roots is essential for predicting the stability and performance of systems, ensuring they operate within safe parameters.