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Answer :
- Multiply the polynomial $f(x)$ by the polynomial $g(x)$.
- Expand the expression by multiplying each term in the first polynomial by each term in the second polynomial.
- Combine like terms to simplify the expression.
- The final result is $(f "." g)(x) = \boxed{49x^4 - 70x^3 + 319x^2 - 420x + 150}$.
### Explanation
1. Understanding the Problem
We are given two polynomials, $f(x) = 7x^3 - 5x^2 + 42x - 30$ and $g(x) = 7x - 5$. Our goal is to find the product of these two polynomials, which is denoted as $(f "." g)(x)$ or $f(x) "." g(x)$. This means we need to multiply each term of $f(x)$ by each term of $g(x)$ and then simplify the resulting expression by combining like terms.
2. Setting up the Multiplication
To find the product $(f "." g)(x)$, we multiply the two polynomials:
$$(f "." g)(x) = (7x^3 - 5x^2 + 42x - 30)(7x - 5)$$
We will expand this expression by multiplying each term in the first polynomial by each term in the second polynomial.
3. Expanding the Expression
Now, we distribute and multiply:
$$(f "." g)(x) = 7x(7x^3 - 5x^2 + 42x - 30) - 5(7x^3 - 5x^2 + 42x - 30)$$
$$= (49x^4 - 35x^3 + 294x^2 - 210x) - (35x^3 - 25x^2 + 210x - 150)$$
$$= 49x^4 - 35x^3 + 294x^2 - 210x - 35x^3 + 25x^2 - 210x + 150$$
4. Combining Like Terms
Next, we combine like terms:
$$(f "." g)(x) = 49x^4 + (-35x^3 - 35x^3) + (294x^2 + 25x^2) + (-210x - 210x) + 150$$
$$= 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
5. Final Answer
Therefore, the product of the two polynomials is:
$$(f "." g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
Comparing this result with the given options, we find that the correct answer is:
$$(f "." g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer graphics. For example, in computer graphics, polynomial multiplication can be used to combine different transformations applied to an object, such as scaling, rotation, and translation. By representing these transformations as polynomials, their combined effect can be efficiently computed using polynomial multiplication.
- Expand the expression by multiplying each term in the first polynomial by each term in the second polynomial.
- Combine like terms to simplify the expression.
- The final result is $(f "." g)(x) = \boxed{49x^4 - 70x^3 + 319x^2 - 420x + 150}$.
### Explanation
1. Understanding the Problem
We are given two polynomials, $f(x) = 7x^3 - 5x^2 + 42x - 30$ and $g(x) = 7x - 5$. Our goal is to find the product of these two polynomials, which is denoted as $(f "." g)(x)$ or $f(x) "." g(x)$. This means we need to multiply each term of $f(x)$ by each term of $g(x)$ and then simplify the resulting expression by combining like terms.
2. Setting up the Multiplication
To find the product $(f "." g)(x)$, we multiply the two polynomials:
$$(f "." g)(x) = (7x^3 - 5x^2 + 42x - 30)(7x - 5)$$
We will expand this expression by multiplying each term in the first polynomial by each term in the second polynomial.
3. Expanding the Expression
Now, we distribute and multiply:
$$(f "." g)(x) = 7x(7x^3 - 5x^2 + 42x - 30) - 5(7x^3 - 5x^2 + 42x - 30)$$
$$= (49x^4 - 35x^3 + 294x^2 - 210x) - (35x^3 - 25x^2 + 210x - 150)$$
$$= 49x^4 - 35x^3 + 294x^2 - 210x - 35x^3 + 25x^2 - 210x + 150$$
4. Combining Like Terms
Next, we combine like terms:
$$(f "." g)(x) = 49x^4 + (-35x^3 - 35x^3) + (294x^2 + 25x^2) + (-210x - 210x) + 150$$
$$= 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
5. Final Answer
Therefore, the product of the two polynomials is:
$$(f "." g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
Comparing this result with the given options, we find that the correct answer is:
$$(f "." g)(x) = 49x^4 - 70x^3 + 319x^2 - 420x + 150$$
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer graphics. For example, in computer graphics, polynomial multiplication can be used to combine different transformations applied to an object, such as scaling, rotation, and translation. By representing these transformations as polynomials, their combined effect can be efficiently computed using polynomial multiplication.
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