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Find \((f \cdot g)(x)\) if \((f(x) = 7x^3 - 5x^2 + 42x - 30)\) and \((g(x) = 7x - 5)\).

A. \((f \cdot g)(x) = 49x^4 - 269x^2 - 150\)

B. \((f \cdot g)(x) = 49x^4 + 269x^2 + 150\)

C. \((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)

D. \((f \cdot g)(x) = 49x^4 + 70x^3 - 319x^2 + 420x - 150\)

Answer :

To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the two polynomial functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].

### Step 1: Identify the Polynomials

The given polynomials are:
- [tex]\(f(x) = 7x^3 - 5x^2 + 42x - 30\)[/tex]
- [tex]\(g(x) = 7x - 5\)[/tex]

### Step 2: Multiply the Polynomials

To calculate [tex]\((f \cdot g)(x)\)[/tex], we'll multiply each term in [tex]\(f(x)\)[/tex] by every term in [tex]\(g(x)\)[/tex].

[tex]\[
\begin{align*}
f(x) \cdot g(x) &= (7x^3 - 5x^2 + 42x - 30) \cdot (7x - 5) \\
&= (7x^3)(7x) + (7x^3)(-5) + (-5x^2)(7x) + (-5x^2)(-5) \\
&\quad + (42x)(7x) + (42x)(-5) + (-30)(7x) + (-30)(-5).
\end{align*}
\][/tex]

### Step 3: Calculate Each Term

We perform the multiplications:

1. [tex]\(7x^3 \cdot 7x = 49x^4\)[/tex]
2. [tex]\(7x^3 \cdot (-5) = -35x^3\)[/tex]
3. [tex]\((-5x^2) \cdot 7x = -35x^3\)[/tex]
4. [tex]\((-5x^2) \cdot (-5) = 25x^2\)[/tex]
5. [tex]\(42x \cdot 7x = 294x^2\)[/tex]
6. [tex]\(42x \cdot (-5) = -210x\)[/tex]
7. [tex]\((-30) \cdot 7x = -210x\)[/tex]
8. [tex]\((-30) \cdot (-5) = 150\)[/tex]

### Step 4: Combine Like Terms

Now, add them together, combining like terms:

- For [tex]\(x^4\)[/tex]: [tex]\(49x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-210x - 210x = -420x\)[/tex]
- Constant Term: [tex]\(150\)[/tex]

The resulting polynomial is:

[tex]\[
49x^4 - 70x^3 + 319x^2 - 420x + 150
\][/tex]

The correct answer is [tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].

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