Question

For which function, defined by a polynomial, are the zeros of the polynomial –4 and –6?

A
y=x2−10x−24y=x^{2}-10x-24y=x
2
−10x−24
B
y=x2+10x+24y=x^2+10x+24y=x
2
+10x+24

C
y=x2+10x−24y=x^2+10x-24y=x
2
+10x−24

D
y=x2−10x+24y=x^2-10x+24y=x
2
−10x+24

Answer 1

To find the polynomial that has zeros at \( -4 \) and \( -6 \), we can use the factored form of a polynomial related to its zeros. Specifically, if \( r_1 \) and \( r_2 \) are the zeros, the polynomial can be expressed as:

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

In this case, the zeros are \( r_1 = -4 \) and \( r_2 = -6 \). Substituting these values into the equation gives:

\[
y = a(x + 4)(x + 6)
\]

Next, we can expand this expression:

\[
y = a[(x + 4)(x + 6)] = a[x^2 + 6x + 4x + 24] = a[x^2 + 10x + 24]
\]

To find the simplest polynomial, we set \( a = 1 \):

\[
y = x^2 + 10x + 24
\]

Now let's check the given options:

- A: \( y = x^2 - 10x - 24 \)
- B: \( y = x^2 + 10x + 24 \)
- C: \( y = x^2 + 10x - 24 \)
- D: \( y = x^2 - 10x + 24 \)

The correct function, defined by a polynomial with zeros at \( -4 \) and \( -6 \), is:

**B: \( y = x^2 + 10x + 24 \)**.