Question: Determine the zeros and the vertical intercept of the polynomial function:
f(x) = 4(x - 5)^2(x + 7)(x + 1)
The zero(s) of the is/are
The horizontal intercept(s) is/are
The vertical intercept is
The degree of the polynomial function is
1. **State the problem:** We need to find the zeros, horizontal intercepts, vertical intercept, and degree of the polynomial function $$f(x) = 4(x - 5)^2(x + 7)(x + 1)$$.
2. **Find the zeros:** Zeros occur where $$f(x) = 0$$.
Set each factor equal to zero:
- $$x - 5 = 0 \Rightarrow x = 5$$ (with multiplicity 2 because of the square)
- $$x + 7 = 0 \Rightarrow x = -7$$
- $$x + 1 = 0 \Rightarrow x = -1$$
So, the zeros are $$x = 5$$ (multiplicity 2), $$x = -7$$, and $$x = -1$$.
3. **Horizontal intercepts:** Horizontal intercepts are the points where the graph crosses the x-axis, which correspond to the zeros.
Thus, the horizontal intercepts are at $$x = 5$$, $$x = -7$$, and $$x = -1$$.
4. **Vertical intercept:** The vertical intercept is the value of $$f(0)$$.
Calculate:
$$f(0) = 4(0 - 5)^2(0 + 7)(0 + 1) = 4( -5)^2(7)(1) = 4 \times 25 \times 7 \times 1 = 700$$
So, the vertical intercept is $$700$$.
5. **Degree of the polynomial:** The degree is the sum of the exponents of the factors.
- $$ (x - 5)^2 $$ contributes degree 2
- $$ (x + 7) $$ contributes degree 1
- $$ (x + 1) $$ contributes degree 1
Total degree = $$2 + 1 + 1 = 4$$.
**Final answers:**
- Zeros: $$x = 5$$ (multiplicity 2), $$x = -7$$, $$x = -1$$
- Horizontal intercepts: $$x = 5$$, $$x = -7$$, $$x = -1$$
- Vertical intercept: $$700$$
- Degree: $$4$$