https://file.aiquickdraw.com/mathbot/file/17266519172308i0qnmhb.jpg

Let's solve the polynomial functions in the image step by step. Each problem asks us to find the zeros, which means solving for \( x \) or \( y \) when the function equals zero.

### 1) \( f(y) = x^2 + 2x - 8 \)

To find the zeros of this quadratic equation, we set \( f(x) = 0 \):
\[
x^2 + 2x - 8 = 0
\]
We can solve this using factoring, completing the square, or the quadratic formula. Let's factor it:

\[
(x + 4)(x - 2) = 0
\]
Thus, \( x = -4 \) and \( x = 2 \) are the zeros.

---

### 2) \( f(y) = y^2 - 1 \)

Set \( f(y) = 0 \):
\[
y^2 - 1 = 0
\]
This is a difference of squares:
\[
(y - 1)(y + 1) = 0
\]
Thus, \( y = 1 \) and \( y = -1 \) are the zeros.

---

### 3) \( f(x) = x^2 + 5x - 14 \)

Set \( f(x) = 0 \):
\[
x^2 + 5x - 14 = 0
\]
Since this does not factor easily, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1 \), \( b = 5 \), and \( c = -14 \):
\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-14)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 56}}{2} = \frac{-5 \pm \sqrt{81}}{2}
\]
\[
x = \frac{-5 \pm 9}{2}
\]
Thus, \( x = \frac{-5 + 9}{2} = 2 \) or \( x = \frac{-5 - 9}{2} = -7 \).

The zeros are \( x = 2 \) and \( x = -7 \).

---

### 4) \( f(x) = x + 6 \)

Set \( f(x) = 0 \):
\[
x + 6 = 0
\]
Thus, \( x = -6 \) is the zero.

---

### 5) \( f(y) = y^2 - 8y + 16 \)

Set \( f(y) = 0 \):
\[
y^2 - 8y + 16 = 0
\]
This is a perfect square trinomial:
\[
(y - 4)^2 = 0
\]
Thus, \( y = 4 \) is the zero.

---

### Summary of Zeros:
1. \( x = -4, 2 \)
2. \( y = 1, -1 \)
3. \( x = 2, -7 \)
4. \( x = -6 \)
5. \( y = 4 \)

Would you like further details or clarifications?

Here are 5 follow-up questions for more practice:

1. How do you factor quadratic equations efficiently?
2. What is the quadratic formula used for?
3. How can you recognize a perfect square trinomial?
4. What is the difference between factoring and using the quadratic formula?
5. How can you solve higher-degree polynomial equations?

**Tip:** Always check if a quadratic can be factored before using the quadratic formula—it saves time!

Solve polynomial equations such as x^2 + 2x - 8 and y^2 - 8y + 16. This tutorial covers solving quadratic equations using factoring, the quadratic formula, and recognizing special forms like perfect square trinomials. Ideal for Grades 8-10.

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