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]
Answer (2)
Polynomials in standard form have terms ordered by decreasing degree.
Examine each polynomial to check the order of the powers of x.
x 4 + 4 x 3 + 10 x 2 has powers in descending order (4, 3, 2).
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 powers of x are 4, 3, and 2. These are in descending order, so this polynomial is in standard form.
x + 4 x 4 + 10 x 2 : The powers of x are 1, 4, and 2. These are not in descending order.
x 4 + 4 x + 10 x 2 : The powers of x are 4, 1, and 2. These are not in descending order.
x 6 + 4 x 3 + 10 x 7 : The powers of x are 6, 3, and 7. These are not in descending order.
Identifying the Correct Polynomial Therefore, the polynomial written in standard form is x 4 + 4 x 3 + 10 x 2 .
Examples
Polynomials are used in many areas of mathematics and science. For example, they are used to model curves and trajectories in physics, such as the path of a projectile. In economics, polynomials can be used to represent cost and revenue functions. Understanding how to write polynomials in standard form helps in simplifying and analyzing these models.
The polynomial written in standard form is x 4 + 4 x 3 + 10 x 2 , which is the only one arranged in decreasing order of degrees. Other options do not follow this order. Therefore, the answer is x 4 + 4 x 3 + 10 x 2 .
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