Can -5 and 12 be the zeroes of polynomial p(x) = 6$$ x^2 $$   - 4x - 170$$ \text{?} $$ Justify your answer.

To determine if $-5$ and $12$ can be the zeroes of the polynomial $p(x) = 6x^2 - 4x - 170$, we need to substitute these values into the polynomial and check if the result is zero. If $p(-5)$ and $p(12)$ both equal zero, then $-5$ and $12$ are indeed the zeroes of the polynomial.

### Step 1: Check $p(-5)$
\[ 
p(-5) = 6(-5)^2 - 4(-5) - 170 
\]

Calculate each term:
\[
(-5)^2 = 25 
\]
\[
6 \times 25 = 150 
\]
\[
-4 \times (-5) = 20 
\]
\[
p(-5) = 150 + 20 - 170 = 170 - 170 = 0
\]

Since $p(-5) = 0$, $-5$ **is** a zero of the polynomial.

### Step 2: Check $p(12)$
\[
p(12) = 6(12)^2 - 4(12) - 170
\]

Calculate each term:
\[
12^2 = 144 
\]
\[
6 \times 144 = 864 
\]
\[
-4 \times 12 = -48 
\]
\[
p(12) = 864 - 48 - 170 = 816 - 170 = 646
\]

Since $p(12) \neq 0$, $12$ is **not** a zero of the polynomial.

### Conclusion:
$-5$ **can** be a zero of the polynomial $p(x) = 6x^2 - 4x - 170$, but $12$ **cannot** be a zero because $p(12)$ does not equal zero.

Would you like more details on this process or have any other questions?

Here are 5 related questions to further explore this concept:

1. How can we find the other zero of the polynomial given that $-5$ is one zero?
2. What is the significance of the discriminant in finding the zeroes of a quadratic polynomial?
3. How do you factorize a quadratic polynomial to find its zeroes?
4. Can a quadratic polynomial have only one zero?
5. What is the geometric interpretation of the zeroes of a polynomial on its graph?

**Tip:** When one zero of a polynomial is known, you can use synthetic division to find the other zero or factorize the polynomial accordingly.

Explore whether -5 and 12 can be the zeroes of the polynomial p(x) = 6x^2 - 4x - 170 through detailed calculation and justification. Understand the application of the quadratic formula and polynomial evaluation. Suitable for advanced high school students.

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