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 first need to simplify the function f(x)
To determine which of the following statements is true, we first need to simplify the function f(x).
Given f(x) = (1/7)x^7 + 12x^6 − x^5 − (15/4)x^4 + (4/3)x^3 + 6x^2.
Statement 1: The degree of f(x) is 7.
The degree of a polynomial is the highest exponent of the variable. In this case, the highest exponent is 7. Therefore, the degree of f(x) is indeed 7. This statement is true.
Statement 2: The leading coefficient of f(x) is 1/7.
The leading coefficient is the coefficient of the term with the highest degree. In this case, the leading coefficient is (1/7). This statement is true.
Statement 3: The constant term of f(x) is 6.
The constant term is the term with no variable, or the term where the exponent is 0. In this case, the constant term is 6x^0, which simplifies to just 6. This statement is true.
Statement 4: f(x) has no x^4 term.
To determine if f(x) has an x^4 term, we need to check if there is any term with x^4. In this case, there is a term with x^4, which is -(15/4)x^4. Therefore, this statement is false.
Statement 5: f(x) is a polynomial function.
A polynomial function is a function that consists of only algebraic terms, where the exponents are whole numbers. In this case, f(x) only contains terms with whole number exponents, so it is a polynomial function. This statement is true.
In summary, statement 1, 2, 3, and 5 are true, while statement 4 is false.
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