Answer :
Sure! Let's address each part of the question step-by-step:
### 1. Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] in the expression. For the polynomial:
[tex]\[ y = 2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60 \][/tex]
The term with the highest power of [tex]\(x\)[/tex] is [tex]\(2x^5\)[/tex]. Therefore, the degree of the polynomial is 5.
### 2. Greatest Term Evaluation
We need to identify the greatest term among the terms [tex]\(2x^5, -5x^4, -30x^3, 5x^2, 88x, 60\)[/tex] for different values of [tex]\(x\)[/tex].
#### a. At [tex]\(x = 0\)[/tex]
Substitute [tex]\(x = 0\)[/tex] into each term:
- [tex]\(2(0)^5 = 0\)[/tex]
- [tex]\(-5(0)^4 = 0\)[/tex]
- [tex]\(-30(0)^3 = 0\)[/tex]
- [tex]\(5(0)^2 = 0\)[/tex]
- [tex]\(88(0) = 0\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 0\)[/tex] is 60.
#### b. At [tex]\(x = 1\)[/tex]
Substitute [tex]\(x = 1\)[/tex] into each term:
- [tex]\(2(1)^5 = 2\)[/tex]
- [tex]\(-5(1)^4 = -5\)[/tex]
- [tex]\(-30(1)^3 = -30\)[/tex]
- [tex]\(5(1)^2 = 5\)[/tex]
- [tex]\(88(1) = 88\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 1\)[/tex] is 88.
#### c. At [tex]\(x = 3\)[/tex]
Substitute [tex]\(x = 3\)[/tex] into each term:
- [tex]\(2(3)^5 = 486\)[/tex]
- [tex]\(-5(3)^4 = -405\)[/tex]
- [tex]\(-30(3)^3 = -810\)[/tex]
- [tex]\(5(3)^2 = 45\)[/tex]
- [tex]\(88(3) = 264\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 3\)[/tex] is 486.
#### d. At [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into each term:
- [tex]\(2(5)^5 = 6250\)[/tex]
- [tex]\(-5(5)^4 = -3125\)[/tex]
- [tex]\(-30(5)^3 = -3750\)[/tex]
- [tex]\(5(5)^2 = 125\)[/tex]
- [tex]\(88(5) = 440\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 5\)[/tex] is 6250.
### 3. Describe the End Behavior of the Polynomial
End behavior describes how a polynomial behaves as [tex]\(x\)[/tex] approaches infinity or negative infinity. For the polynomial [tex]\(y = 2x^5\)[/tex], since the term [tex]\(2x^5\)[/tex] has the highest degree, it dominates the end behavior.
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(+\infty\)[/tex]), the term [tex]\(2x^5\)[/tex] causes the polynomial to go to positive infinity. So, the polynomial goes to infinity.
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]), the term [tex]\(2x^5\)[/tex] causes the polynomial to go to negative infinity. So, the polynomial goes to negative infinity.
In summary:
- Degree: 5
- Greatest terms for [tex]\(x = 0, 1, 3, 5\)[/tex]: 60, 88, 486, 6250
- End behavior: infinity as [tex]\(x \to +\infty\)[/tex] and negative infinity as [tex]\(x \to -\infty\)[/tex]
### 1. Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of the variable [tex]\(x\)[/tex] in the expression. For the polynomial:
[tex]\[ y = 2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60 \][/tex]
The term with the highest power of [tex]\(x\)[/tex] is [tex]\(2x^5\)[/tex]. Therefore, the degree of the polynomial is 5.
### 2. Greatest Term Evaluation
We need to identify the greatest term among the terms [tex]\(2x^5, -5x^4, -30x^3, 5x^2, 88x, 60\)[/tex] for different values of [tex]\(x\)[/tex].
#### a. At [tex]\(x = 0\)[/tex]
Substitute [tex]\(x = 0\)[/tex] into each term:
- [tex]\(2(0)^5 = 0\)[/tex]
- [tex]\(-5(0)^4 = 0\)[/tex]
- [tex]\(-30(0)^3 = 0\)[/tex]
- [tex]\(5(0)^2 = 0\)[/tex]
- [tex]\(88(0) = 0\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 0\)[/tex] is 60.
#### b. At [tex]\(x = 1\)[/tex]
Substitute [tex]\(x = 1\)[/tex] into each term:
- [tex]\(2(1)^5 = 2\)[/tex]
- [tex]\(-5(1)^4 = -5\)[/tex]
- [tex]\(-30(1)^3 = -30\)[/tex]
- [tex]\(5(1)^2 = 5\)[/tex]
- [tex]\(88(1) = 88\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 1\)[/tex] is 88.
#### c. At [tex]\(x = 3\)[/tex]
Substitute [tex]\(x = 3\)[/tex] into each term:
- [tex]\(2(3)^5 = 486\)[/tex]
- [tex]\(-5(3)^4 = -405\)[/tex]
- [tex]\(-30(3)^3 = -810\)[/tex]
- [tex]\(5(3)^2 = 45\)[/tex]
- [tex]\(88(3) = 264\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 3\)[/tex] is 486.
#### d. At [tex]\(x = 5\)[/tex]
Substitute [tex]\(x = 5\)[/tex] into each term:
- [tex]\(2(5)^5 = 6250\)[/tex]
- [tex]\(-5(5)^4 = -3125\)[/tex]
- [tex]\(-30(5)^3 = -3750\)[/tex]
- [tex]\(5(5)^2 = 125\)[/tex]
- [tex]\(88(5) = 440\)[/tex]
- [tex]\(60 = 60\)[/tex]
The greatest term when [tex]\(x = 5\)[/tex] is 6250.
### 3. Describe the End Behavior of the Polynomial
End behavior describes how a polynomial behaves as [tex]\(x\)[/tex] approaches infinity or negative infinity. For the polynomial [tex]\(y = 2x^5\)[/tex], since the term [tex]\(2x^5\)[/tex] has the highest degree, it dominates the end behavior.
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(+\infty\)[/tex]), the term [tex]\(2x^5\)[/tex] causes the polynomial to go to positive infinity. So, the polynomial goes to infinity.
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]), the term [tex]\(2x^5\)[/tex] causes the polynomial to go to negative infinity. So, the polynomial goes to negative infinity.
In summary:
- Degree: 5
- Greatest terms for [tex]\(x = 0, 1, 3, 5\)[/tex]: 60, 88, 486, 6250
- End behavior: infinity as [tex]\(x \to +\infty\)[/tex] and negative infinity as [tex]\(x \to -\infty\)[/tex]
