To find the resulting polynomial \( f(x) \cdot g(x) \) where:
- \( f(x) = 8x - 2 \)
- \( g(x) = -10x^2 - x - 7 \)
we will multiply these two polynomials:
\[ f(x) \cdot g(x) = (8x - 2)(-10x^2 - x - 7) \]
Now we will distribute \( f(x) \) over \( g(x) \):
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Multiply \( 8x \) by each term in \( g(x) \):
- \( 8x \cdot (-10x^2) = -80x^3 \)
- \( 8x \cdot (-x) = -8x^2 \)
- \( 8x \cdot (-7) = -56x \)
-
Multiply \( -2 \) by each term in \( g(x) \):
- \( -2 \cdot (-10x^2) = 20x^2 \)
- \( -2 \cdot (-x) = 2x \)
- \( -2 \cdot (-7) = 14 \)
Now we combine all these results together:
\[ f(x) \cdot g(x) = -80x^3 + (-8x^2) + (-56x) + 20x^2 + 2x + 14 \]
Next, we can simplify by combining like terms:
-
For the \( x^2 \) terms:
- \( -8x^2 + 20x^2 = 12x^2 \)
-
For the \( x \) terms:
- \( -56x + 2x = -54x \)
Putting it all back together, we have:
\[ f(x) \cdot g(x) = -80x^3 + 12x^2 - 54x + 14 \]
Thus, the resulting polynomial \( f(x) \cdot g(x) \) is:
\[ \boxed{-80x^3 + 12x^2 - 54x + 14} \]