Let's analyze the function [tex]\( h(x) = x^3 - 4x^2 - 3x + 18 \)[/tex] to determine the nature and number of its roots.
1. Identify the type of polynomial:
The given function is a cubic polynomial of the form [tex]\( ax^3 + bx^2 + cx + d \)[/tex].
2. Use the Fundamental Theorem of Algebra:
According to the Fundamental Theorem of Algebra, a polynomial of degree [tex]\( n \)[/tex] has exactly [tex]\( n \)[/tex] roots (real or complex), counting multiplicities. Therefore, a cubic polynomial such as [tex]\( h(x) \)[/tex] will have exactly 3 roots in total.
3. Determine the nature of the roots:
The roots of a cubic polynomial can be real or complex. To classify them:
- Real Roots: Roots that are real numbers.
- Complex Roots: Roots that are complex numbers, which will always occur in conjugate pairs.
4. Check the real number solutions for [tex]\( h(x) \)[/tex]:
We have found the real number solutions to the polynomial equation [tex]\( h(x) = 0 \)[/tex]. The real roots found are [tex]\( -2 \)[/tex] and [tex]\( 3 \)[/tex].
5. Count the real roots:
From the determined roots, we can see there are 2 distinct real roots: [tex]\( -2 \)[/tex] and [tex]\( 3 \)[/tex].
Since there are 2 real roots, the best description of the roots of the function [tex]\( h(x) = x^3 - 4x^2 - 3x + 18 \)[/tex] is:
2 real number solutions
Therefore, the correct option is:
- 2 real number solutions.
