For which function, defined by a polynomial, are the zeros of the polynomial –8 and 2?

Question

A
y=x2+10x+16y=x^2+10x+16y=x
2
+10x+16

B
y=x2−6x−16y=x^2-6x-16y=x
2
−6x−16

C
y=x2+6x−16y=x^2+6x-16y=x
2
+6x−16

D
y=x2−6x+16y=x^2-6x+16y=x
2
−6x+16

Answer 1

To find the polynomial with the specified zeros -8 and 2, we can use the fact that if a polynomial has zeros at \( r_1 \) and \( r_2 \), it can be expressed in factored form as:

\[ y = k(x - r_1)(x - r_2) \]

In this case, the zeros are \( r_1 = -8 \) and \( r_2 = 2 \). Therefore, we can write:

\[ y = k(x + 8)(x - 2) \]

For simplicity, we can let \( k = 1 \):

\[ y = (x + 8)(x - 2) \]

Now, let's expand this:

\[ y = x^2 - 2x + 8x - 16 \] \[ y = x^2 + 6x - 16 \]

So, the polynomial we derived is

\[ y = x^2 + 6x - 16 \]

This corresponds to option C:

C: \( y = x^2 + 6x - 16 \)