Please find your answer bellow and please follow me for the same
Step-by-step explanation:
One way to create a polynomial function whose graph never crosses the horizontal axis is to ensure that all the coefficients of the terms with odd powers are zero. This way, the function will not have any real roots.
For example, let's create a 4th-degree polynomial function:
\[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \]
To ensure that the function never crosses the horizontal axis, we can set the coefficients of the terms with odd powers (i.e., \( bx^3 \) and \( dx \)) to zero:
\[ f(x) = ax^4 + 0x^3 + cx^2 + 0x + e \]
This simplifies to:
\[ f(x) = ax^4 + cx^2 + e \]
We can choose any non-zero values for \( a \), \( c \), and \( e \) to define the function.
For example, let's choose \( a = 1 \), \( c = 2 \), and \( e = 3 \):
\[ f(x) = x^4 + 2x^2 + 3 \]
This polynomial function is a 4th-degree polynomial that never crosses the horizontal axis. Its graph will have a shape determined by the values of \( a \), \( c \), and \( e \), but it will always remain above or below the horizontal axis, never crossing it.
