Answer :
Sure! Let's solve the problem step-by-step.
We are given the polynomial [tex]\( y = 2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60 \)[/tex].
### 1. Identify the degree of the polynomial.
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the polynomial.
In this polynomial, the highest power of [tex]\( x \)[/tex] is 5 (from the term [tex]\( 2x^5 \)[/tex]).
Therefore, the degree of the polynomial is 5.
### 2. Determine which term is greatest for different values of [tex]\( x \)[/tex].
Let's evaluate the polynomial terms to see which one is greatest when:
#### a. [tex]\( x = 0 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 0^5 = 0 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 0^4 = 0 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 0^3 = 0 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 0^2 = 0 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 0 = 0 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 0 \)[/tex] is [tex]\( 60 \)[/tex] with the value being 60.
#### b. [tex]\( x = 1 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 1^5 = 2 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 1^4 = -5 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 1^3 = -30 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 1^2 = 5 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 1 = 88 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 1 \)[/tex] is [tex]\( 88x \)[/tex] with the value being 88.
#### c. [tex]\( x = 3 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 3^5 = 486 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 3^4 = -405 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 3^3 = -810 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 3^2 = 45 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 3 = 264 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 3 \)[/tex] is [tex]\( 2x^5 \)[/tex] with the value being 486.
#### d. [tex]\( x = 5 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 5^5 = 6250 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 5^4 = -3125 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 5^3 = -3750 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 5^2 = 125 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 5 = 440 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 5 \)[/tex] is [tex]\( 2x^5 \)[/tex] with the value being 6250.
### 3. Describe the end behavior of the polynomial.
To understand the end behavior, we look at the term with the highest degree (the leading term), which is [tex]\( 2x^5 \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( 2x^5 \)[/tex] will become increasingly large in the positive direction, so [tex]\( y \)[/tex] will approach [tex]\( +\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( 2x^5 \)[/tex] will become increasingly large in the negative direction (since raising a negative number to an odd power results in a negative number), so [tex]\( y \)[/tex] will approach [tex]\( -\infty \)[/tex].
Thus, the end behavior of the polynomial is:
- [tex]\( y \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]
- [tex]\( y \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]
In summary:
1. The degree of the polynomial is 5.
2. The greatest term for each specific value of [tex]\( x \)[/tex] is:
- At [tex]\( x = 0 \)[/tex]: [tex]\( 60 \)[/tex] with value 60
- At [tex]\( x = 1 \)[/tex]: [tex]\( 88x \)[/tex] with value 88
- At [tex]\( x = 3 \)[/tex]: [tex]\( 2x^5 \)[/tex] with value 486
- At [tex]\( x = 5 \)[/tex]: [tex]\( 2x^5 \)[/tex] with value 6250
3. The end behavior of the polynomial is:
- [tex]\( y \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]
- [tex]\( y \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]
We are given the polynomial [tex]\( y = 2x^5 - 5x^4 - 30x^3 + 5x^2 + 88x + 60 \)[/tex].
### 1. Identify the degree of the polynomial.
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the polynomial.
In this polynomial, the highest power of [tex]\( x \)[/tex] is 5 (from the term [tex]\( 2x^5 \)[/tex]).
Therefore, the degree of the polynomial is 5.
### 2. Determine which term is greatest for different values of [tex]\( x \)[/tex].
Let's evaluate the polynomial terms to see which one is greatest when:
#### a. [tex]\( x = 0 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 0^5 = 0 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 0^4 = 0 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 0^3 = 0 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 0^2 = 0 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 0 = 0 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 0 \)[/tex] is [tex]\( 60 \)[/tex] with the value being 60.
#### b. [tex]\( x = 1 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 1^5 = 2 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 1^4 = -5 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 1^3 = -30 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 1^2 = 5 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 1 = 88 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 1 \)[/tex] is [tex]\( 88x \)[/tex] with the value being 88.
#### c. [tex]\( x = 3 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 3^5 = 486 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 3^4 = -405 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 3^3 = -810 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 3^2 = 45 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 3 = 264 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 3 \)[/tex] is [tex]\( 2x^5 \)[/tex] with the value being 486.
#### d. [tex]\( x = 5 \)[/tex]
The polynomial terms are:
- [tex]\( 2x^5 \)[/tex] becomes [tex]\( 2 \cdot 5^5 = 6250 \)[/tex]
- [tex]\( -5x^4 \)[/tex] becomes [tex]\( -5 \cdot 5^4 = -3125 \)[/tex]
- [tex]\( -30x^3 \)[/tex] becomes [tex]\( -30 \cdot 5^3 = -3750 \)[/tex]
- [tex]\( 5x^2 \)[/tex] becomes [tex]\( 5 \cdot 5^2 = 125 \)[/tex]
- [tex]\( 88x \)[/tex] becomes [tex]\( 88 \cdot 5 = 440 \)[/tex]
- The constant term is [tex]\( 60 \)[/tex]
The greatest term when [tex]\( x = 5 \)[/tex] is [tex]\( 2x^5 \)[/tex] with the value being 6250.
### 3. Describe the end behavior of the polynomial.
To understand the end behavior, we look at the term with the highest degree (the leading term), which is [tex]\( 2x^5 \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( 2x^5 \)[/tex] will become increasingly large in the positive direction, so [tex]\( y \)[/tex] will approach [tex]\( +\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( 2x^5 \)[/tex] will become increasingly large in the negative direction (since raising a negative number to an odd power results in a negative number), so [tex]\( y \)[/tex] will approach [tex]\( -\infty \)[/tex].
Thus, the end behavior of the polynomial is:
- [tex]\( y \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]
- [tex]\( y \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]
In summary:
1. The degree of the polynomial is 5.
2. The greatest term for each specific value of [tex]\( x \)[/tex] is:
- At [tex]\( x = 0 \)[/tex]: [tex]\( 60 \)[/tex] with value 60
- At [tex]\( x = 1 \)[/tex]: [tex]\( 88x \)[/tex] with value 88
- At [tex]\( x = 3 \)[/tex]: [tex]\( 2x^5 \)[/tex] with value 486
- At [tex]\( x = 5 \)[/tex]: [tex]\( 2x^5 \)[/tex] with value 6250
3. The end behavior of the polynomial is:
- [tex]\( y \)[/tex] approaches [tex]\( +\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]
- [tex]\( y \)[/tex] approaches [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]
