Choose the polynomial written in standard form.
[tex]x^4+4 x^3+10 x^2[/tex]
[tex]x+4 x^4+10 x^2[/tex]
[tex]x^4+4 x+10 x^2[/tex]
[tex]x^6+4 x^3+10 x^7[/tex]

A polynomial in standard form has terms ordered from highest to lowest degree.
Examine each polynomial to check if the exponents decrease from left to right.
x 4 + 4 x 3 + 10 x 2 is the only polynomial with terms in decreasing order of degree.
The polynomial in standard form is x 4 + 4 x 3 + 10 x 2 ​ .

Explanation

Understanding Standard Form A polynomial in standard form is written with the term with the highest degree first and decreasing to the term with the lowest degree. We need to identify which of the given polynomials is written in this form.

Checking Each Polynomial Let's examine each option:

x 4 + 4 x 3 + 10 x 2 : The exponents are 4, 3, and 2. This is in decreasing order, so this polynomial is in standard form.

x + 4 x 4 + 10 x 2 : The exponents are 1, 4, and 2. This is not in decreasing order.

x 4 + 4 x + 10 x 2 : The exponents are 4, 1, and 2. This is not in decreasing order.

x 6 + 4 x 3 + 10 x 7 : The exponents are 6, 3, and 7. This is not in decreasing order.

Identifying the Correct Polynomial Therefore, the polynomial written in standard form is x 4 + 4 x 3 + 10 x 2 .


Examples
Understanding polynomial forms is essential in various fields, such as physics and engineering, where equations often involve polynomials. For example, when analyzing the trajectory of a projectile, the height can be modeled as a polynomial function of time. Writing the polynomial in standard form helps in easily identifying the degree and leading coefficient, which are crucial for understanding the behavior of the projectile's motion. Similarly, in signal processing, polynomials are used to represent filters, and the standard form aids in analyzing the filter's characteristics.

Answered by GinnyAnswer | 2025-07-03

The polynomial in standard form is x 4 + 4 x 3 + 10 x 2 , as it arranges the terms in decreasing order of their degrees. The other options do not follow this ordering, making them not in standard form.
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Answered by Anonymous | 2025-07-04