Answer :
To determine whether the given statement is true or false, let's analyze the properties of polynomials with odd degrees.
1. Definition of Polynomial Degree:
- The degree of a polynomial is the highest power of the variable in the polynomial expression.
2. Odd-Degree Polynomials:
- Polynomials of odd degrees (like [tex]\( x^3, x^5, \)[/tex] etc.) have some specific characteristics. The most important for this question is their end behavior.
3. End Behavior:
- For polynomials of odd degrees, as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], the polynomial value will also approach [tex]\( +\infty \)[/tex]. Conversely, as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], the value of the polynomial will approach [tex]\( -\infty \)[/tex]. This indicates that the function is unbounded in both the positive and negative directions.
4. Roots of Odd-Degree Polynomials:
- Another critical aspect is that odd-degree polynomials will always have at least one real root. This means that the graph of the polynomial will intersect the x-axis at least once.
Taking these facts into consideration:
- Since an odd-degree polynomial approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases without bound and [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] decreases without bound,
- And since it will have at least one real root allowing it to cross from negative to positive values (or vice versa),
We can conclude that:
- The range of an odd-degree polynomial indeed extends from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Therefore, the statement "The range of a polynomial with odd degree will be [tex]\( (-\infty, \infty) \)[/tex]" is True.
1. Definition of Polynomial Degree:
- The degree of a polynomial is the highest power of the variable in the polynomial expression.
2. Odd-Degree Polynomials:
- Polynomials of odd degrees (like [tex]\( x^3, x^5, \)[/tex] etc.) have some specific characteristics. The most important for this question is their end behavior.
3. End Behavior:
- For polynomials of odd degrees, as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], the polynomial value will also approach [tex]\( +\infty \)[/tex]. Conversely, as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], the value of the polynomial will approach [tex]\( -\infty \)[/tex]. This indicates that the function is unbounded in both the positive and negative directions.
4. Roots of Odd-Degree Polynomials:
- Another critical aspect is that odd-degree polynomials will always have at least one real root. This means that the graph of the polynomial will intersect the x-axis at least once.
Taking these facts into consideration:
- Since an odd-degree polynomial approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] increases without bound and [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] decreases without bound,
- And since it will have at least one real root allowing it to cross from negative to positive values (or vice versa),
We can conclude that:
- The range of an odd-degree polynomial indeed extends from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Therefore, the statement "The range of a polynomial with odd degree will be [tex]\( (-\infty, \infty) \)[/tex]" is True.
