Answer :
Sure, let's find the product of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] step-by-step.
We are given:
[tex]\[ f(x) = 5x^2 \][/tex]
[tex]\[ g(x) = x^3 + 2x^2 - 5x \][/tex]
To find [tex]\( f(x) \cdot g(x) \)[/tex], we multiply these two functions together:
[tex]\[ f(x) \cdot g(x) = (5x^2) \cdot (x^3 + 2x^2 - 5x) \][/tex]
Let's distribute [tex]\( 5x^2 \)[/tex] across each term inside the parentheses:
1. [tex]\( 5x^2 \cdot x^3 = 5x^{2+3} = 5x^5 \)[/tex]
2. [tex]\( 5x^2 \cdot 2x^2 = 5 \cdot 2 \cdot x^{2+2} = 10x^4 \)[/tex]
3. [tex]\( 5x^2 \cdot (-5x) = 5 \cdot (-5) \cdot x^{2+1} = -25x^3 \)[/tex]
Now, we combine these results:
[tex]\[ f(x) \cdot g(x) = 5x^5 + 10x^4 - 25x^3 \][/tex]
So, the product [tex]\( f(x) \cdot g(x) \)[/tex] is:
[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]
Now let's compare this with the options provided:
- [tex]\( x^3 + 7x^2 - 5 \)[/tex]
- [tex]\( 5x^6 + 10x^4 - 25x^2 \)[/tex]
- [tex]\( 5x^5 + 10x^4 - 25x^3 \)[/tex]
- [tex]\( -x^3 + 3x^2 + 5x \)[/tex]
The correct option is:
[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]
We are given:
[tex]\[ f(x) = 5x^2 \][/tex]
[tex]\[ g(x) = x^3 + 2x^2 - 5x \][/tex]
To find [tex]\( f(x) \cdot g(x) \)[/tex], we multiply these two functions together:
[tex]\[ f(x) \cdot g(x) = (5x^2) \cdot (x^3 + 2x^2 - 5x) \][/tex]
Let's distribute [tex]\( 5x^2 \)[/tex] across each term inside the parentheses:
1. [tex]\( 5x^2 \cdot x^3 = 5x^{2+3} = 5x^5 \)[/tex]
2. [tex]\( 5x^2 \cdot 2x^2 = 5 \cdot 2 \cdot x^{2+2} = 10x^4 \)[/tex]
3. [tex]\( 5x^2 \cdot (-5x) = 5 \cdot (-5) \cdot x^{2+1} = -25x^3 \)[/tex]
Now, we combine these results:
[tex]\[ f(x) \cdot g(x) = 5x^5 + 10x^4 - 25x^3 \][/tex]
So, the product [tex]\( f(x) \cdot g(x) \)[/tex] is:
[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]
Now let's compare this with the options provided:
- [tex]\( x^3 + 7x^2 - 5 \)[/tex]
- [tex]\( 5x^6 + 10x^4 - 25x^2 \)[/tex]
- [tex]\( 5x^5 + 10x^4 - 25x^3 \)[/tex]
- [tex]\( -x^3 + 3x^2 + 5x \)[/tex]
The correct option is:
[tex]\[ \boxed{5x^5 + 10x^4 - 25x^3} \][/tex]
