0. Functions

Common Functions

0. Functions

Common Functions

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Multiple Choice

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

Polynomial with $n=3,a_{n}=2$

Polynomial with $n=2,a_{n}=3$

Polynomial with $n=2,a_{n}=2$

Not a polynomial function.

Verified step by step guidance

Step 1: Recall the definition of a polynomial function. A polynomial function is an expression of the form $ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 $, where $ n $ is a non-negative integer, and the coefficients $ a_n, a_{n-1}, \dots, a_0 $ are real numbers.

Step 2: Analyze the given function $ f(x) = 3x^2 + 5x + 2 $. Check if it fits the form of a polynomial. Here, the terms $ 3x^2 $, $ 5x $, and $ 2 $ all have non-negative integer exponents (2, 1, and 0, respectively), and the coefficients (3, 5, and 2) are real numbers. Therefore, this is a polynomial function.

Step 3: Write the polynomial in standard form. A polynomial is in standard form when its terms are arranged in descending order of their exponents. The given function $ f(x) = 3x^2 + 5x + 2 $ is already in standard form.

Step 4: Identify the degree of the polynomial. The degree of a polynomial is the highest power of $ x $ in the expression. In this case, the highest power of $ x $ is 2, so the degree is $ n = 2 $.

Step 5: Determine the leading coefficient. The leading coefficient is the coefficient of the term with the highest power of $ x $. Here, the term with the highest power is $ 3x^2 $, so the leading coefficient is $ a_n = 3 $.