(2x+5)(x-3)^2(8x+1)^5 answer is at most 8 why

Answer 1

To analyze the expression $(2x+5)(x-3{)}^2(8x+1{)}^5$ and understand why the answer is at most 8, we can look at the asymptotic behavior as well as the individual components of the expression.

1. Identify Degree of Each Factor:

   • The term $(2x+5)$ is linear, so its degree is 1.
   • The term $(x-3{)}^2$ is quadratic, so its degree is 2.
   • The term $(8x+1{)}^5$ is a polynomial of degree 5.

2. Total Degree:

   • Therefore, the total degree of the entire expression is: $$1+2+5=8.$$ This confirms that the expression will behave like a polynomial of degree 8 for large values of $x$.

3. Evaluating Limits:

   • To evaluate the behavior as $x$ approaches infinity, consider the leading terms.

   • As $x\to \mathrm{\infty }$:

     • $2x+5\approx 2x$
     • $(x-3{)}^2\approx x^2$
     • $(8x+1{)}^5\approx (8x{)}^5=8^5{x}^5=32768x^5$

   • Hence: $$(2x+5)(x-3{)}^2(8x+1{)}^5\approx (2x)(x^2)(32768x^5)=65536x^8.$$

   • Thus, as $x$ approaches infinity, the expression grows polynomially without bound.

4. Saturation Condition:

   • However, if we are given that the output is a bounded value, we might have constraints on $x$ leading the resulting polynomial's output to be maximized at a certain finite point not located at $x=\mathrm{\infty }$.

5. Finding Critical Points:

   • To find the maximum value of the expression over a finite interval, we would typically set the derivative of the expression equal to zero and solve for $x$, or evaluate it at its boundary values to determine local maxima/minima.

6. Behavior Constraints:

   • To accurately state that the answer is at most 8, it could imply:
     1. The expression may have local maximum restrictions.
     2. There may be specific $x$ values (such as roots or turning points) that limit the output to this range.

In conclusion, without specifying the constraints or bounds for $x$, the expression could grow without bound, but if we are given that the answer is at most 8, then either through calculus or constraints imposed, it leads us to a finite maximum.