To find \( f(x) \cdot g(x) \), we need to first recall the expressions for \( f(x) \) and \( g(x) \):
\[ f(x) = 5x - 2 \]
\[ g(x) = 10x^2 - 3x - 7 \]
Now, we'll multiply \( f(x) \) and \( g(x) \):
\[ f(x) \cdot g(x) = (5x - 2)(10x^2 - 3x - 7) \]
We'll distribute \( (5x - 2) \) across \( (10x^2 - 3x - 7) \).
-
Multiply \( 5x \) by each term in \( g(x) \):
\[ 5x \cdot 10x^2 = 50x^3 \] \[ 5x \cdot (-3x) = -15x^2 \] \[ 5x \cdot (-7) = -35x \]
-
Multiply \( -2 \) by each term in \( g(x) \):
\[ -2 \cdot 10x^2 = -20x^2 \] \[ -2 \cdot (-3x) = 6x \] \[ -2 \cdot (-7) = 14 \]
Now, we can combine all the terms together:
\[ f(x) \cdot g(x) = 50x^3 + (-15x^2) + (-35x) + (-20x^2) + 6x + 14 \]
Next, we combine like terms:
- Cubic term: \( 50x^3 \)
- Quadratic terms: \( -15x^2 - 20x^2 = -35x^2 \)
- Linear terms: \( -35x + 6x = -29x \)
- Constant term: \( 14 \)
Putting it all together, we have:
\[ f(x) \cdot g(x) = 50x^3 - 35x^2 - 29x + 14 \]
Therefore, the resulting polynomial is:
\[ \boxed{50x^3 - 35x^2 - 29x + 14} \]