To find the value of [tex]\( (f \cdot g)(x) \)[/tex] when [tex]\( f(x) = 0.5x^2 - 2 \)[/tex] and [tex]\( g(x) = 8x^3 + 2 \)[/tex], we need to follow these steps:
1. Identify the functions:
- [tex]\( f(x) = 0.5x^2 - 2 \)[/tex]
- [tex]\( g(x) = 8x^3 + 2 \)[/tex]
2. Calculate the product [tex]\( (f \cdot g)(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ = (0.5x^2 - 2) \cdot (8x^3 + 2) \][/tex]
3. Expand the product:
- Expand each term in [tex]\( f(x) \)[/tex] and multiply it by each term in [tex]\( g(x) \)[/tex].
[tex]\[
(0.5x^2 - 2) \cdot (8x^3 + 2) = (0.5x^2 \cdot 8x^3) + (0.5x^2 \cdot 2) + (-2 \cdot 8x^3) + (-2 \cdot 2)
\][/tex]
4. Multiply the terms:
[tex]\[
= (0.5 \cdot 8)x^{2+3} + (0.5 \cdot 2)x^2 + (-2 \cdot 8)x^3 + (-2 \cdot 2)
\][/tex]
[tex]\[
= 4x^5 + x^2 - 16x^3 - 4
\][/tex]
5. Rearrange into standard polynomial form:
[tex]\[
= 4x^5 - 16x^3 + x^2 - 4
\][/tex]
6. Identify the coefficients:
- Coefficient of [tex]\( x^5 \)[/tex] is [tex]\( 4 \)[/tex]
- Coefficient of [tex]\( x^3 \)[/tex] is [tex]\( -16 \)[/tex]
- Coefficient of [tex]\( x^2 \)[/tex] is [tex]\( 1 \)[/tex]
- Constant term is [tex]\( -4 \)[/tex]
Based on these calculations, the correct values to fill the blanks in the expression [tex]\((f \cdot g)(x) = \boxed{x^5} - \boxed{x^3} + \boxed{x^2} - \boxed{}\)[/tex] are:
- First blank for [tex]\( x^5 \)[/tex] term: [tex]\(\boxed{4}\)[/tex]
- Second blank for [tex]\( x^3 \)[/tex] term: [tex]\(\boxed{16}\)[/tex]
- Third blank for [tex]\( x^2 \)[/tex] term: [tex]\(\boxed{1}\)[/tex]
- Fourth blank for the constant term: [tex]\(\boxed{4}\)[/tex]
Thus, the completed expression is:
[tex]\[ (f \cdot g)(x) = 4x^5 - 16x^3 + x^2 - 4. \][/tex]
