Alright, let's analyze the given polynomial function step-by-step to find its degree and the y-intercept.
### Step 1: Find the Degree of the Polynomial
The polynomial function is given as:
[tex]\[ f(x) = (x + 4)^2 (x - 1)^5 (x - 2) \][/tex]
To find the degree of the polynomial, we need to determine the highest power of [tex]\( x \)[/tex] when the polynomial is expanded. The degree is calculated by adding the exponents of [tex]\( x \)[/tex] in each term of the factors:
- The factor [tex]\((x + 4)^2\)[/tex] contributes a degree of 2.
- The factor [tex]\((x - 1)^5\)[/tex] contributes a degree of 5.
- The factor [tex]\((x - 2)\)[/tex] contributes a degree of 1.
So the total degree of the polynomial is:
[tex]\[ 2 + 5 + 1 = 8 \][/tex]
Thus, the degree of the polynomial [tex]\( f(x) \)[/tex] is 8.
### Step 2: Find the y-Intercept
The y-intercept of a polynomial is the value of the function when [tex]\( x = 0 \)[/tex]. To find this, we substitute [tex]\( x = 0 \)[/tex] into the polynomial:
[tex]\[ f(0) = (0 + 4)^2 (0 - 1)^5 (0 - 2) \][/tex]
Simplifying each factor:
- [tex]\((0 + 4)^2 = 4^2 = 16\)[/tex]
- [tex]\((0 - 1)^5 = (-1)^5 = -1\)[/tex]
- [tex]\((0 - 2) = -2\)[/tex]
Multiplying these values together:
[tex]\[ f(0) = 16 \times -1 \times -2 = 16 \times 2 = 32 \][/tex]
Thus, the y-intercept is 32.
### Step 3: Express the Intercept as an Ordered Pair
The y-intercept is the point at which the graph of the function crosses the y-axis, i.e., where [tex]\( x = 0 \)[/tex]. This can be expressed as the ordered pair:
[tex]\[ (0, 32) \][/tex]
### Final Answer
So, the solutions are:
- The degree of the polynomial [tex]\( f(x) \)[/tex] is 8.
- The y-intercept, expressed as an ordered pair, is [tex]\( (0, 32) \)[/tex].
This completes the required calculations for finding the degree and the y-intercept of the polynomial function.
