Can -5 and 12 be the zeros of the polynomial PX is equal to 6X squared minus minus 4X -170

To determine whether \(-5\) and \(12\) are the zeros of the polynomial \(P(X) = 6X^2 - 4X - 170\), we need to check if substituting these values into the polynomial yields \(0\). A zero of a polynomial satisfies \(P(X) = 0\).

### Step 1: Test \(X = -5\)

Substitute \(X = -5\) into the polynomial:
\[
P(-5) = 6(-5)^2 - 4(-5) - 170
\]
Simplifying:
\[
P(-5) = 6(25) + 20 - 170 = 150 + 20 - 170 = 0
\]
Thus, \(-5\) is a zero of the polynomial.

### Step 2: Test \(X = 12\)

Substitute \(X = 12\) into the polynomial:
\[
P(12) = 6(12)^2 - 4(12) - 170
\]
Simplifying:
\[
P(12) = 6(144) - 48 - 170 = 864 - 48 - 170 = 646
\]
Since \(P(12) = 646\), \(12\) is **not** a zero of the polynomial.

### Conclusion:
\(-5\) is a zero of the polynomial, but \(12\) is not.

Would you like further details or have any other questions?

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Here are five related questions:
1. How do you find the zeros of a polynomial in general?
2. Can a polynomial have more than two real zeros?
3. What methods can be used to factor quadratic polynomials?
4. How can you check if a complex number is a zero of a polynomial?
5. How does the degree of a polynomial relate to the number of possible zeros?

**Tip:** Always double-check each calculation when testing possible zeros to avoid errors!

Can -5 and 12 be the zeros of the polynomial P(X) = 6X^2 - 4X - 170?

Check whether -5 and 12 are zeros of the quadratic polynomial P(X) = 6X^2 - 4X - 170 with a detailed step-by-step evaluation. This problem explores key algebra concepts and polynomial evaluation. Suitable for students in Grades 8-10.

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