Answer :
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply the given functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
1. Write down the functions:
[tex]\(f(x) = 7x^3 - 5x^2 + 42x - 30\)[/tex]
[tex]\(g(x) = 7x - 5\)[/tex]
2. Multiply the polynomials:
When multiplying two polynomials, each term in the first polynomial has to be multiplied by each term in the second polynomial. This process involves:
- Multiplying [tex]\(7x^3\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(-5x^2\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(42x\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(-30\)[/tex] by [tex]\(7x - 5\)[/tex]
3. Perform the multiplication step-by-step:
- [tex]\(7x^3 \cdot 7x = 49x^4\)[/tex]
- [tex]\(7x^3 \cdot (-5) = -35x^3\)[/tex]
- [tex]\(-5x^2 \cdot 7x = -35x^3\)[/tex]
- [tex]\(-5x^2 \cdot (-5) = 25x^2\)[/tex]
- [tex]\(42x \cdot 7x = 294x^2\)[/tex]
- [tex]\(42x \cdot (-5) = -210x\)[/tex]
- [tex]\(-30 \cdot 7x = -210x\)[/tex]
- [tex]\(-30 \cdot (-5) = 150\)[/tex]
4. Combine the terms:
Collect all the like terms:
- [tex]\(49x^4\)[/tex] (only one term for [tex]\(x^4\)[/tex])
- [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- [tex]\(-210x - 210x = -420x\)[/tex]
- [tex]\(150\)[/tex]
5. Resulting polynomial:
Putting it all together, the product [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex]
Therefore, the correct answer is [tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].
1. Write down the functions:
[tex]\(f(x) = 7x^3 - 5x^2 + 42x - 30\)[/tex]
[tex]\(g(x) = 7x - 5\)[/tex]
2. Multiply the polynomials:
When multiplying two polynomials, each term in the first polynomial has to be multiplied by each term in the second polynomial. This process involves:
- Multiplying [tex]\(7x^3\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(-5x^2\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(42x\)[/tex] by [tex]\(7x - 5\)[/tex]
- Multiplying [tex]\(-30\)[/tex] by [tex]\(7x - 5\)[/tex]
3. Perform the multiplication step-by-step:
- [tex]\(7x^3 \cdot 7x = 49x^4\)[/tex]
- [tex]\(7x^3 \cdot (-5) = -35x^3\)[/tex]
- [tex]\(-5x^2 \cdot 7x = -35x^3\)[/tex]
- [tex]\(-5x^2 \cdot (-5) = 25x^2\)[/tex]
- [tex]\(42x \cdot 7x = 294x^2\)[/tex]
- [tex]\(42x \cdot (-5) = -210x\)[/tex]
- [tex]\(-30 \cdot 7x = -210x\)[/tex]
- [tex]\(-30 \cdot (-5) = 150\)[/tex]
4. Combine the terms:
Collect all the like terms:
- [tex]\(49x^4\)[/tex] (only one term for [tex]\(x^4\)[/tex])
- [tex]\(-35x^3 - 35x^3 = -70x^3\)[/tex]
- [tex]\(25x^2 + 294x^2 = 319x^2\)[/tex]
- [tex]\(-210x - 210x = -420x\)[/tex]
- [tex]\(150\)[/tex]
5. Resulting polynomial:
Putting it all together, the product [tex]\((f \cdot g)(x)\)[/tex] is:
[tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex]
Therefore, the correct answer is [tex]\((f \cdot g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150\)[/tex].
