Which polynomial functions are written in standard form? Select all that apply.
[tex]$f(x)=8-x^5$[/tex]
[tex]$f(x)=-3 x^5+5 x-2$[/tex]
[tex]$f(x)=2 x^5+2 x+x^3$[/tex]
[tex]$f(x)=x^3-8 x^2$[/tex]
Answer (1)
A polynomial function is in standard form when its terms are arranged in descending order of their exponents.
Rewrite f ( x ) = 8 − x 5 as f ( x ) = − x 5 + 8 , which is in standard form.
The function f ( x ) = − 3 x 5 + 5 x − 2 is in standard form.
Rewrite f ( x ) = 2 x 5 + 2 x + x 3 as f ( x ) = 2 x 5 + x 3 + 2 x , which is in standard form.
The function f ( x ) = x 3 − 8 x 2 is in standard form.
All given functions are in standard form.
Explanation
Understanding Standard Form A polynomial function is in standard form when its terms are arranged in descending order of their exponents. We need to check each given function to see if it meets this criterion.
Analyzing the First Function Let's examine the first function, f ( x ) = 8 − x 5 . To determine if it's in standard form, we rewrite it with the term having the highest exponent first: f ( x ) = − x 5 + 8 . This is now in standard form because the exponent of x decreases from 5 to 0.
Analyzing the Second Function Next, consider the second function, f ( x ) = − 3 x 5 + 5 x − 2 . The exponents are 5, 1, and 0, which are in descending order. Thus, this function is in standard form.
Analyzing the Third Function Now, let's analyze the third function, f ( x ) = 2 x 5 + 2 x + x 3 . The exponents are 5, 1, and 3. To check if it's in standard form, we need to rearrange the terms in descending order of exponents: f ( x ) = 2 x 5 + x 3 + 2 x . This is now in standard form.
Analyzing the Fourth Function Finally, let's examine the fourth function, f ( x ) = x 3 − 8 x 2 . The exponents are 3 and 2, which are in descending order. Therefore, this function is in standard form.
Final Answer In conclusion, all the given polynomial functions are written in standard form.
Examples
Understanding polynomial functions and their standard form is crucial in various fields, such as physics and engineering. For instance, when modeling the trajectory of a projectile, the height of the projectile can be described by a quadratic polynomial function in standard form, h ( t ) = a t 2 + b t + c , where a , b , and c are constants, and t represents time. By analyzing the coefficients and the order of the terms, engineers can predict the maximum height and range of the projectile. Similarly, in signal processing, polynomial functions are used to approximate complex signals, and expressing them in standard form simplifies analysis and manipulation.
