Answer :
Sure, I can help break down the steps to answer this question about 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 expression. In this polynomial, the term with the highest exponent is [tex]\( 2x^5 \)[/tex]. So, the degree of the polynomial is 5.
### 2. Determine the Greatest Term for Specific Values of [tex]\( x \)[/tex]
To find out which term is the greatest when substituting different values for [tex]\( x \)[/tex], we'll evaluate each term separately and compare their values.
a. When [tex]\( x = 0 \)[/tex]:
- [tex]\( 2x^5 = 2(0)^5 = 0 \)[/tex]
- [tex]\( -5x^4 = -5(0)^4 = 0 \)[/tex]
- [tex]\( -30x^3 = -30(0)^3 = 0 \)[/tex]
- [tex]\( 5x^2 = 5(0)^2 = 0 \)[/tex]
- [tex]\( 88x = 88(0) = 0 \)[/tex]
- Constant term = 60
At [tex]\( x = 0 \)[/tex], the greatest term is 60.
b. When [tex]\( x = 1 \)[/tex]:
- [tex]\( 2x^5 = 2(1)^5 = 2 \)[/tex]
- [tex]\( -5x^4 = -5(1)^4 = -5 \)[/tex]
- [tex]\( -30x^3 = -30(1)^3 = -30 \)[/tex]
- [tex]\( 5x^2 = 5(1)^2 = 5 \)[/tex]
- [tex]\( 88x = 88(1) = 88 \)[/tex]
- Constant term = 60
At [tex]\( x = 1 \)[/tex], the greatest term is 88x, which evaluates to 88.
c. When [tex]\( x = 3 \)[/tex]:
- [tex]\( 2x^5 = 2(3)^5 = 486 \)[/tex]
- [tex]\( -5x^4 = -5(3)^4 = -405 \)[/tex]
- [tex]\( -30x^3 = -30(3)^3 = -810 \)[/tex]
- [tex]\( 5x^2 = 5(3)^2 = 45 \)[/tex]
- [tex]\( 88x = 88(3) = 264 \)[/tex]
- Constant term = 60
At [tex]\( x = 3 \)[/tex], the greatest term is 2x^5, which evaluates to 486.
d. When [tex]\( x = 5 \)[/tex]:
- [tex]\( 2x^5 = 2(5)^5 = 6250 \)[/tex]
- [tex]\( -5x^4 = -5(5)^4 = -3125 \)[/tex]
- [tex]\( -30x^3 = -30(5)^3 = -3750 \)[/tex]
- [tex]\( 5x^2 = 5(5)^2 = 125 \)[/tex]
- [tex]\( 88x = 88(5) = 440 \)[/tex]
- Constant term = 60
At [tex]\( x = 5 \)[/tex], the greatest term is 2x^5, which evaluates to 6250.
### 3. Describe the End Behavior of the Polynomial
The end behavior of a polynomial is determined by its leading term, which is the term with the highest degree. In this polynomial, the leading term is [tex]\( 2x^5 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, the term [tex]\( 2x^5 \)[/tex] becomes very large, leading the polynomial to approach positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( 2x^5 \)[/tex] (which is positive) when raised to an odd power (5), results in a negative value, hence the polynomial approaches negative infinity.
Therefore, the end behavior can be described as follows:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] approaches infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] approaches negative infinity.
I hope this helps you understand the polynomial and its characteristics! If you have any more questions, feel free to ask.
### 1. Identify the Degree of the Polynomial
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the expression. In this polynomial, the term with the highest exponent is [tex]\( 2x^5 \)[/tex]. So, the degree of the polynomial is 5.
### 2. Determine the Greatest Term for Specific Values of [tex]\( x \)[/tex]
To find out which term is the greatest when substituting different values for [tex]\( x \)[/tex], we'll evaluate each term separately and compare their values.
a. When [tex]\( x = 0 \)[/tex]:
- [tex]\( 2x^5 = 2(0)^5 = 0 \)[/tex]
- [tex]\( -5x^4 = -5(0)^4 = 0 \)[/tex]
- [tex]\( -30x^3 = -30(0)^3 = 0 \)[/tex]
- [tex]\( 5x^2 = 5(0)^2 = 0 \)[/tex]
- [tex]\( 88x = 88(0) = 0 \)[/tex]
- Constant term = 60
At [tex]\( x = 0 \)[/tex], the greatest term is 60.
b. When [tex]\( x = 1 \)[/tex]:
- [tex]\( 2x^5 = 2(1)^5 = 2 \)[/tex]
- [tex]\( -5x^4 = -5(1)^4 = -5 \)[/tex]
- [tex]\( -30x^3 = -30(1)^3 = -30 \)[/tex]
- [tex]\( 5x^2 = 5(1)^2 = 5 \)[/tex]
- [tex]\( 88x = 88(1) = 88 \)[/tex]
- Constant term = 60
At [tex]\( x = 1 \)[/tex], the greatest term is 88x, which evaluates to 88.
c. When [tex]\( x = 3 \)[/tex]:
- [tex]\( 2x^5 = 2(3)^5 = 486 \)[/tex]
- [tex]\( -5x^4 = -5(3)^4 = -405 \)[/tex]
- [tex]\( -30x^3 = -30(3)^3 = -810 \)[/tex]
- [tex]\( 5x^2 = 5(3)^2 = 45 \)[/tex]
- [tex]\( 88x = 88(3) = 264 \)[/tex]
- Constant term = 60
At [tex]\( x = 3 \)[/tex], the greatest term is 2x^5, which evaluates to 486.
d. When [tex]\( x = 5 \)[/tex]:
- [tex]\( 2x^5 = 2(5)^5 = 6250 \)[/tex]
- [tex]\( -5x^4 = -5(5)^4 = -3125 \)[/tex]
- [tex]\( -30x^3 = -30(5)^3 = -3750 \)[/tex]
- [tex]\( 5x^2 = 5(5)^2 = 125 \)[/tex]
- [tex]\( 88x = 88(5) = 440 \)[/tex]
- Constant term = 60
At [tex]\( x = 5 \)[/tex], the greatest term is 2x^5, which evaluates to 6250.
### 3. Describe the End Behavior of the Polynomial
The end behavior of a polynomial is determined by its leading term, which is the term with the highest degree. In this polynomial, the leading term is [tex]\( 2x^5 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, the term [tex]\( 2x^5 \)[/tex] becomes very large, leading the polynomial to approach positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( 2x^5 \)[/tex] (which is positive) when raised to an odd power (5), results in a negative value, hence the polynomial approaches negative infinity.
Therefore, the end behavior can be described as follows:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( y \)[/tex] approaches infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] approaches negative infinity.
I hope this helps you understand the polynomial and its characteristics! If you have any more questions, feel free to ask.
