Write Function: To determine if the function f(x) is even, odd, or neither, we need to evaluate f(-x) and compare it to f(x). If f(-x) = f(x), the function is even. If f(-x) = -f(x), the function is odd. If neither condition is met, the function is neither even nor odd. First, let's write down the original function: f(x) = 5x^3 - 35x^2 - 3x + 21Substitute -x: Now, let's substitute -x for x in the function to find f(-x): f(-x) = 5(-x)^3 - 35(-x)^2 - 3(-x) + 21Simplify f(-x): Next, we simplify f(-x) by calculating the powers of -x: f(-x) = 5(-x)^3 - 35(-x)^2 - 3(-x) + 21 f(-x) = 5(-1)^3x^3 - 35(-1)^2x^2 + 3x + 21 f(-x) = -5x^3 - 35x^2 + 3x + 21Compare f(-x) with f(x): Now we compare f(-x) with f(x): f(x) = 5x^3 - 35x^2 - 3x + 21 f(-x) = -5x^3 - 35x^2 + 3x + 21 We can see that f(-x) is not equal to f(x) and also not equal to -f(x), because the signs of the x^3 and x terms are different.Determine Even/Odd/Neither: Since f(-x) is neither equal to f(x) nor -f(x), the function f(x) = 5x^3 - 35x^2 - 3x + 21 is neither even nor odd.

Algebra 2

Divide polynomials using long division

Full solution

5 m^{3}-35 m^{2}-3 m+21

Write Function: To determine if the function $f(x)$ is even, odd, or neither, we need to evaluate $f(-x)$ and compare it to $f(x)$ . If $f(-x) = f(x)$ , the function is even. If $f(-x) = -f(x)$ , the function is odd. If neither condition is met, the function is neither even nor odd. $\newline$ First, let's write down the original function: $\newline$ $f(x) = 5x^3 - 35x^2 - 3x + 21$
Substitute $-x$ : Now, let's substitute $-x$ for $x$ in the function to find $f(-x)$ : $f(-x) = 5(-x)^3 - 35(-x)^2 - 3(-x) + 21$
Simplify $f(-x)$ : Next, we simplify $f(-x)$ by calculating the powers of $-x$ :
$f(-x) = 5(-x)^3 - 35(-x)^2 - 3(-x) + 21$
$f(-x) = 5(-1)^3x^3 - 35(-1)^2x^2 + 3x + 21$
$f(-x) = -5x^3 - 35x^2 + 3x + 21$
Compare $f(-x)$ with $f(x)$ : Now we compare $f(-x)$ with $f(x)$ :
$f(x) = 5x^3 - 35x^2 - 3x + 21$
$f(-x) = -5x^3 - 35x^2 + 3x + 21$
We can see that $f(-x)$ is not equal to $f(x)$ and also not equal to $-f(x)$ , because the signs of the $x^3$ and $f(x)$ $0$ terms are different.
Determine Even/Odd/Neither: Since $f(-x)$ is neither equal to $f(x)$ nor $-f(x)$ , the function $f(x) = 5x^3 - 35x^2 - 3x + 21$ is neither even nor odd.