Determine the resulting polynomial:

Question

f, of, x, equals, 5, x, minus, 3
f(x)=5x−3
g, of, x, equals, minus, 4, x, squared, plus, 4, x, plus, 7
g(x)=−4x
2
+4x+7
Find: , f, of, x, dot, g, of, x
Find: f(x)⋅g(x)

Answer 1

To find \( f(x) \cdot g(x) \), we first need to express both functions clearly:

1. \( f(x) = 5x - 3 \)
2. \( g(x) = -4x^2 + 4x + 7 \)

Now we can multiply these two functions:

\[ f(x) \cdot g(x) = (5x - 3)(-4x^2 + 4x + 7) \]

We will use the distributive property (also known as the FOIL method here) to expand this product:

1. Multiply \(5x\) with each term in \(g(x)\):

   • \(5x \cdot (-4x^2) = -20x^3\)
   • \(5x \cdot 4x = 20x^2\)
   • \(5x \cdot 7 = 35x\)

2. Multiply \(-3\) with each term in \(g(x)\):

   • \(-3 \cdot (-4x^2) = 12x^2\)
   • \(-3 \cdot 4x = -12x\)
   • \(-3 \cdot 7 = -21\)

Now, we can combine all these results together:

\[ f(x) \cdot g(x) = -20x^3 + 20x^2 + 35x + 12x^2 - 12x - 21 \]

Next, combine like terms:

• The \(x^2\) terms: \(20x^2 + 12x^2 = 32x^2\)
• The \(x\) terms: \(35x - 12x = 23x\)

So the combined polynomial is:

\[ f(x) \cdot g(x) = -20x^3 + 32x^2 + 23x - 21 \]

Thus, the resulting polynomial is:

\[ \boxed{-20x^3 + 32x^2 + 23x - 21} \]