Part 1Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. f(x)equals=22xplus+x cubedx3 Part 1Determine whether f(x) is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. It is not a polynomial because the variable x is raised to the [input]enter your response here power, which is not a nonnegative integer.(Type an integer or a fraction.) B. It is a polynomial of degree [input]enter your response here .(Type an integer or a fraction.) C. It is not a polynomial because the constant term is absent.

Question

Part 1Determine whether the following function is a polynomial function. If the function is a polynomial​ function, state its degree. If it is​ not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. ​f(x)equals=22xplus+x cubedx3 Part 1Determine whether​ f(x) is a polynomial or not. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. It is not a polynomial because the variable x is raised to the [input]enter your response here ​power, which is not a nonnegative integer.​(Type an integer or a​ fraction.) B. It is a polynomial of degree [input]enter your response here .​(Type an integer or a​ fraction.) C. It is not a polynomial because the constant term is absent.

BlackTom AI · Accepted Answer

To start, restate what the function is and what it means to be a polynomial. A polynomial is a finite sum of terms with nonnegative integer powers of the variable, each multiplied by a coefficient. If the function can be written in standard form as a0 + a1 x + a2 x^2 + ... + an x^n with n a nonnegative integer, it is a polynomial, its degree is n, the leading term is a_n x^n, and the constant term is a0.

Option A: 'It is not a polynomial because the variable x is raised to the [input] power, which is not a nonnegative integer.' In this case, the exponents seen in the function are 1 (from 22x) and 3 (from x^3), both nonnegative integers. There is no exponent that is negative or fractional, so this statement is inaccurate. The function can indeed be written as x^3 + 22x, which fits the polynomial form.

Option B: 'It is a polynomial of degree [input]'. From the standard form x^3 + 22x, the highest exponent is 3, so the degree is 3. Therefore, this statement is correct because the degree matches the largest exponent in the polynomial, and the expression is a finite sum of powers with nonnegative integer exponents.

Option C: 'It is not a polynomial because the constant term is absent.' The absence of a constant term (i.e., the constant term being 0) does not disqualify a function from being a polynomial. A polynomial can have a0 = 0 and still be perfectly valid. The missing constant term is simply a0 = 0, not evidence that the function is non-polynomial.

Additional clarifications: The standard form of the polynomial is written with terms ordered by decreasing powers of x: x^3 + 22x. The leading term is x^3, which determines the leading coefficient (1) and the degree (3). The constant term is 0, reflecting that there is no constant term present in the expression.

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