Question

Write a polynomial function in standard form with the given zeros: x = 2, -3, -7. a) f(x) = (x - 2)(x + 3)(x + 7) b) f(x) = (x + 2)(x - 3)(x - 7) c) f(x) = (x - 2)(x - 3)(x - 7) d) f(x) = (x + 2)(x + 3)(x + 7)

Answer 1

The correct option that represents a polynomial function in standard form with the given zeros x = 2, -3, -7 is:

a)
`\( f(x) = (x - 2)(x + 3)(x + 7) \)`

To write a polynomial function with given zeros x = 2, -3, -7 , we need to use the fact that if x = a is a zero of a polynomial, then (x - a) is a factor of that polynomial. Let's apply this to each option:

Step 1: Apply the Zero-Factor Principle

The zero-factor principle tells us that if
`\( x = 2 \), then \( (x - 2) \) is a factor; if \( x = -3 \), then \( (x + 3) \) is a factor (since \( x - (-3) = x + 3 \)); and if \( x = -7 \), then \( (x + 7) \) is a factor (since \( x - (-7) = x + 7 \)).`

Step 2: Form the Polynomial Function

A polynomial with these zeros will be the product of these factors. Thus, the polynomial is:

`\[ f(x) = (x - 2)(x + 3)(x + 7) \]`

Step 3: Match with the Given Options

Now, let's match this polynomial with the given options:

- a)
`\( f(x) = (x - 2)(x + 3)(x + 7) \) -` This matches our polynomial.

- b)
`\( f(x) = (x + 2)(x - 3)(x - 7) \)` - This does not match our zeros.

- c)
`\( f(x) = (x - 2)(x - 3)(x - 7) \)` - This does not match our zeros.

- d)
`\( f(x) = (x + 2)(x + 3)(x + 7) \)` - This does not match our zeros.