To analyze the behavior of the polynomial [tex]\( f(x) = 4x^8 - 2x^7 - 3x^4 + 5x^2 + 1 \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity, let's examine the dominant term in the polynomial.
The polynomial is:
[tex]\[ f(x) = 4x^8 - 2x^7 - 3x^4 + 5x^2 + 1 \][/tex]
### As [tex]\( x \)[/tex] approaches positive infinity:
1. We note that the highest degree term in the polynomial is [tex]\( 4x^8 \)[/tex].
2. As [tex]\( x \to \infty \)[/tex], the term [tex]\( 4x^8 \)[/tex] will dominate all other terms because it grows much faster than the other terms.
3. The leading term [tex]\( 4x^8 \)[/tex] is positive, and as [tex]\( x \)[/tex] increases without bound, [tex]\( 4x^8 \)[/tex] will also increase without bound.
Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex]. In other words, [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] approaches positive infinity.
### As [tex]\( x \)[/tex] approaches negative infinity:
1. Similarly, we consider the highest degree term [tex]\( 4x^8 \)[/tex].
2. Notice that [tex]\( 4x^8 \)[/tex] is still positive for negative values of [tex]\( x \)[/tex], because any negative number raised to an even power becomes positive.
3. All other terms will be much smaller in magnitude compared to [tex]\( 4x^8 \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex]. In other words, [tex]\( y \)[/tex] increases as [tex]\( x \)[/tex] approaches negative infinity as well.
Based on the above analysis, we summarize the behavior of the polynomial:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] increases.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] increases.
Hence, the statements:
- "As [tex]\( x \)[/tex] approaches negative infinity [tex]\( y \)[/tex] increases" is true.
- "As [tex]\( x \)[/tex] approaches positive infinity [tex]\( y \)[/tex] decreases" is false.
- "As [tex]\( x \)[/tex] approaches positive infinity [tex]\( y \)[/tex] approaches 0" is false.
