Question 1 Of 5Select The Correct Answer.Consider Functions { F $}$ And { G $} . . . { \begin{align} f(x) &= 5x^4 + X^3 - 2x^2 + 4, \\ g(x) &= 5x^3 - 5x^2 + 7x - 4. \end{align} \} Which Expression Is Equal To [$ F(x)

Mar 11, 2025 by ADMIN 220 views

Question 1 of 5: Select the Correct Answer

Consider Functions f(x) and g(x)

Problem Description

We are given two functions, f(x) and g(x), and we need to find an expression that is equal to f(x). The functions are defined as:

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

${ g(x) = 5x^3 - 5x^2 + 7x - 4 }$

Step 1: Analyze the Functions

To find an expression equal to f(x), we need to analyze the given functions and identify any relationships between them.

Step 2: Identify the Relationship Between f(x) and g(x)

Upon closer inspection, we can see that g(x) is a polynomial function of degree 3, while f(x) is a polynomial function of degree 4. However, we can try to express f(x) in terms of g(x) by using algebraic manipulations.

Step 3: Express f(x) in Terms of g(x)

Let's try to express f(x) in terms of g(x) by using the following approach:

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

${ g(x) = 5x^3 - 5x^2 + 7x - 4 }$

We can try to factor out a common term from f(x) and g(x) to see if we can express f(x) in terms of g(x).

Step 4: Factor Out a Common Term

Let's try to factor out a common term from f(x) and g(x):

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

${ g(x) = 5x^3 - 5x^2 + 7x - 4 }$

We can see that both functions have a common term of x^3. Let's try to factor out x^3 from both functions:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

${ g(x) = 5x^3 - 5x^2 + 7x - 4 }$

Now, we can see that f(x) can be expressed in terms of g(x) by using the following expression:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

Step 5: Simplify the Expression

We can simplify the expression by combining like terms:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

Conclusion

We have found an expression that is equal to f(x), which is:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

This expression is equal to f(x) and can be used to represent f(x) in terms of g(x).

The Final Answer is:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

Discussion Category: Mathematics

This problem involves algebraic manipulations and polynomial functions. The solution requires analyzing the given functions, identifying relationships between them, and using algebraic manipulations to express f(x) in terms of g(x). The final answer is a polynomial expression that is equal to f(x).
Question 2 of 5: Select the Correct Answer

Consider Functions f(x) and g(x)

Problem Description

We are given two functions, f(x) and g(x), and we need to find an expression that is equal to f(x). The functions are defined as:

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

${ g(x) = 5x^3 - 5x^2 + 7x - 4 }$

Q&A Session

Q: What is the relationship between f(x) and g(x)?

A: The relationship between f(x) and g(x) is that g(x) is a polynomial function of degree 3, while f(x) is a polynomial function of degree 4.

Q: How can we express f(x) in terms of g(x)?

A: We can express f(x) in terms of g(x) by using algebraic manipulations. We can try to factor out a common term from f(x) and g(x) to see if we can express f(x) in terms of g(x).

Q: What is the common term that we can factor out from f(x) and g(x)?

A: The common term that we can factor out from f(x) and g(x) is x^3.

Q: How can we simplify the expression f(x) = x^3(5x + 1) - 2x^2 + 4?

A: We can simplify the expression by combining like terms. The simplified expression is:

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

Q: What is the final answer to the problem?

A: The final answer to the problem is:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

Additional Questions

Q: What is the degree of the polynomial function f(x)?

A: The degree of the polynomial function f(x) is 4.

Q: What is the degree of the polynomial function g(x)?

A: The degree of the polynomial function g(x) is 3.

Q: Can we express g(x) in terms of f(x)?

A: No, we cannot express g(x) in terms of f(x) using the given information.

Conclusion

We have found an expression that is equal to f(x), which is:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

This expression is equal to f(x) and can be used to represent f(x) in terms of g(x).

The Final Answer is:

${ f(x) = x^3(5x + 1) - 2x^2 + 4 }$

Discussion Category: Mathematics

Question 3 of 5: Select the Correct Answer

Consider Functions f(x) and g(x)

Problem Description

We are given two functions, f(x) and g(x), and we need to find the value of f(x) when x = 2.

Q&A Session

Q: What is the value of f(x) when x = 2?

A: To find the value of f(x) when x = 2, we need to substitute x = 2 into the expression for f(x).

Q: What is the expression for f(x)?

A: The expression for f(x) is:

${ f(x) = 5x^4 + x^3 - 2x^2 + 4 }$

Q: How can we substitute x = 2 into the expression for f(x)?

A: We can substitute x = 2 into the expression for f(x) by replacing x with 2 in the expression.

Q: What is the value of f(2)?

A: To find the value of f(2), we need to evaluate the expression:

${ f(2) = 5(2)^4 + (2)^3 - 2(2)^2 + 4 }$

Q: How can we evaluate the expression f(2)?

A: We can evaluate the expression f(2) by following the order of operations (PEMDAS):

Evaluate the exponents: (2)^4 = 16, (2)^3 = 8, (2)^2 = 4
Multiply 5 by 16: 5(16) = 80
Add 8: 80 + 8 = 88
Subtract 8: 88 - 8 = 80
Add 4: 80 + 4 = 84

Q: What is the final answer to the problem?

A: The final answer to the problem is:

${ f(2) = 84 }$

Conclusion

We have found the value of f(x) when x = 2, which is:

${ f(2) = 84 }$

This value is equal to f(2) and can be used to represent f(2) in terms of the expression for f(x).

The Final Answer is:

${ f(2) = 84 }$

Discussion Category: Mathematics

This problem involves evaluating a polynomial expression at a specific value of x. The solution requires substituting x = 2 into the expression for f(x) and evaluating the resulting expression using the order of operations (PEMDAS). The final answer is a numerical value that represents f(2).