Answer :
Sure, let's go through the solution to this polynomial question step-by-step:
1. Degree of the Polynomial:
- The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] with a non-zero coefficient. In the given function [tex]\( f(x) = -45x^5 + 264x^4 - 512x^3 + 422x^2 - 147x + 18 \)[/tex], the highest power of [tex]\( x \)[/tex] is 5. Therefore, the degree of [tex]\( f \)[/tex] is 5.
2. Leading Coefficient:
- The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the term with the highest degree is [tex]\(-45x^5\)[/tex], so the leading coefficient is [tex]\(-45\)[/tex].
3. End Behavior:
- To determine the end behavior, we examine the leading term [tex]\(-45x^5\)[/tex]:
- Right hand end behavior: As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]), the dominant term is [tex]\(-45x^5\)[/tex]. Since the coefficient [tex]\(-45\)[/tex] is negative and the degree is odd, [tex]\( f(x) \)[/tex] will tend to [tex]\(-\infty\)[/tex].
- Left hand end behavior: As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the dominant term, being negative and of odd degree, makes [tex]\( f(x) \)[/tex] tend to [tex]\(\infty\)[/tex].
4. Zeros of the Polynomial:
- The zeros of a polynomial are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. From the given answer, the zeros of [tex]\( f(x) \)[/tex] are approximately [tex]\( x = 3, 1.2, 1, \)[/tex] and two repeated roots at approximately [tex]\( x = 0.333 \)[/tex].
To summarize:
- The degree of the polynomial is 5.
- The leading coefficient is [tex]\(-45\)[/tex].
- The right-hand end behavior as [tex]\( x \to \infty \)[/tex] is [tex]\( f(x) \to -\infty \)[/tex].
- The left-hand end behavior as [tex]\( x \to -\infty \)[/tex] is [tex]\( f(x) \to \infty \)[/tex].
- The zeros of the polynomial are approximately [tex]\( 3, 1.2, 1, \)[/tex] and two at approximately [tex]\( 0.333 \)[/tex].
1. Degree of the Polynomial:
- The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] with a non-zero coefficient. In the given function [tex]\( f(x) = -45x^5 + 264x^4 - 512x^3 + 422x^2 - 147x + 18 \)[/tex], the highest power of [tex]\( x \)[/tex] is 5. Therefore, the degree of [tex]\( f \)[/tex] is 5.
2. Leading Coefficient:
- The leading coefficient is the coefficient of the term with the highest degree. In our polynomial, the term with the highest degree is [tex]\(-45x^5\)[/tex], so the leading coefficient is [tex]\(-45\)[/tex].
3. End Behavior:
- To determine the end behavior, we examine the leading term [tex]\(-45x^5\)[/tex]:
- Right hand end behavior: As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]), the dominant term is [tex]\(-45x^5\)[/tex]. Since the coefficient [tex]\(-45\)[/tex] is negative and the degree is odd, [tex]\( f(x) \)[/tex] will tend to [tex]\(-\infty\)[/tex].
- Left hand end behavior: As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the dominant term, being negative and of odd degree, makes [tex]\( f(x) \)[/tex] tend to [tex]\(\infty\)[/tex].
4. Zeros of the Polynomial:
- The zeros of a polynomial are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. From the given answer, the zeros of [tex]\( f(x) \)[/tex] are approximately [tex]\( x = 3, 1.2, 1, \)[/tex] and two repeated roots at approximately [tex]\( x = 0.333 \)[/tex].
To summarize:
- The degree of the polynomial is 5.
- The leading coefficient is [tex]\(-45\)[/tex].
- The right-hand end behavior as [tex]\( x \to \infty \)[/tex] is [tex]\( f(x) \to -\infty \)[/tex].
- The left-hand end behavior as [tex]\( x \to -\infty \)[/tex] is [tex]\( f(x) \to \infty \)[/tex].
- The zeros of the polynomial are approximately [tex]\( 3, 1.2, 1, \)[/tex] and two at approximately [tex]\( 0.333 \)[/tex].
