Can-5 and 12 be the zeroes of polynomial p(x) = 6x ^ 2 - 4x - 1707 ) Justify your answer.

To determine if \(-5\) and \(12\) can be the zeroes of the polynomial \(p(x) = 6x^2 - 4x - 1707\), 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 \(p(-5)\)
\[ 
p(-5) = 6(-5)^2 - 4(-5) - 1707 
\]

Calculate each term:
\[
(-5)^2 = 25 
\]
\[
6 \times 25 = 150 
\]
\[
-4 \times (-5) = 20 
\]
\[
p(-5) = 150 + 20 - 1707 = 170 - 1707 = -1537
\]

Since \(p(-5) \neq 0\), \(-5\) is **not** a zero of the polynomial.

### Step 2: Check \(p(12)\)
\[
p(12) = 6(12)^2 - 4(12) - 1707
\]

Calculate each term:
\[
12^2 = 144 
\]
\[
6 \times 144 = 864 
\]
\[
-4 \times 12 = -48 
\]
\[
p(12) = 864 - 48 - 1707 = 816 - 1707 = -891
\]

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

### Conclusion:
Neither \(-5\) nor \(12\) are zeroes of the polynomial \(p(x) = 6x^2 - 4x - 1707\), as substituting them into the polynomial does not result in 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. What are the actual zeroes of the polynomial \(p(x) = 6x^2 - 4x - 1707\)?
2. How can we find the zeroes of a quadratic polynomial using the quadratic formula?
3. What does it mean for a value to be a zero of a polynomial?
4. Can a quadratic polynomial have more than two zeroes?
5. How do we interpret the graph of a polynomial in relation to its zeroes?

**Tip:** When checking if a value is a zero of a polynomial, always substitute the value into the polynomial and simplify. If the result is zero, the value is indeed a zero of the polynomial.

Explore whether -5 and 12 can be the zeroes of the polynomial p(x) = 6x^2 - 4x - 1707 through detailed substitution and analysis. This problem involves quadratic equations and polynomial evaluation. Understand why these values do not satisfy the polynomial equation.

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