Question

Refer to the following function for the questions that follow.

p(x) = 1x² + 3x - 4x² + 6x⁴ - 1
Determine if the function is a polynomial or not. If it is not, explain why. If it is a polynomial, classify the polynomial, identify the degree, identify the leading coefficient, and write it in standard form.

Answer 1

p(x) = 1x² + 3x - 4x² + 6x⁴ - 1 is indeed a polynomial, specifically a 4th-degree quartic polynomial with a leading coefficient of 6, and its standard form is 6x⁴ - 3x² + 3x - 1.

Step-by-step explanation:

The function in question is p(x) = 1x² + 3x - 4x² + 6x⁴ - 1. To determine whether p(x) is a polynomial, we should look for exponents of the variable x that are non-negative integers and for coefficients that are real numbers.

In this case, all exponents on x (2, 1, 4) are non-negative integers, and the coefficients (1, 3, -4, 6, -1) are all real numbers. Thus, p(x) is a polynomial. Now we need to classify it by degree and write it in standard form, where the terms are ordered from highest degree to lowest degree.

When combining like terms, we find that p(x) simplifies to 6x⁴ - 3x² + 3x - 1. This is the polynomial written in standard form. The highest degree term is 6x⁴, which makes this a 4th-degree polynomial, also known as a quartic polynomial. The coefficient of the highest degree term, 6, is the leading coefficient.