Evaluating Polynomials | Determining True Statements about a Given Function

Let f be the function given by f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2. Which of the following statements is true?

To determine which of the following statements is true, we need to evaluate the given function and compare the results

To determine which of the following statements is true, we need to evaluate the given function and compare the results.

The function is given by:

f(x) = (1/7)x^7 + 12x^6 – x^5 – (15/4)x^4 + (4/3)x^3 + 6x^2

First, let’s write down the given options:

A. The leading term of f(x) is x^7.
B. The degree of f(x) is 6.
C. The constant term of f(x) is 6.
D. The coefficient of the x^3 term is 4/3.

Now, let’s evaluate each statement.

A. The leading term of f(x) is x^7.
The leading term of a polynomial function is the term with the highest degree. The highest degree in the given function is 7, so the leading term is indeed x^7. Therefore, statement A is true.

B. The degree of f(x) is 6.
The degree of a polynomial function is the highest exponent of the variable. In this case, the highest exponent is 7, so the degree of f(x) is 7, not 6. Therefore, statement B is false.

C. The constant term of f(x) is 6.
The constant term is the term that does not have any variable attached to it. In this function, the term with no variable is indeed 6. Therefore, statement C is true.

D. The coefficient of the x^3 term is 4/3.
The coefficient of a term is the number multiplied by the variable. For the x^3 term, the coefficient is (4/3). Therefore, statement D is true.

To summarize:
– Statement A is true.
– Statement B is false.
– Statement C is true.
– Statement D is true.

Therefore, the correct statement is: “The leading term of f(x) is x^7, the constant term of f(x) is 6, and the coefficient of the x^3 term is 4/3.”

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