If one zero of the polynomial is 7 and product of zeroes is -35, then polynomial is:

Options

A. `x^2 +12 x-35`
B. `x^2 -12 x-35`
C. `-x^2 +12 x-35`
D. `x^2 +12 x-35`

Answer

To find the polynomial given one zero and the product of the zeroes, we can follow these steps:

### Step 1: Identify the known values
We know that one zero (let's call it α) is 7, and the product of the zeroes (α * β) is -35.

### Step 2: Find the other zero
Let the other zero be β. We can use the product of the zeroes to find β:
\[ α * β = -35 \]
Substituting α = 7:
\[ 7 * β = -35 \]
To find β, divide both sides by 7:
\[ β = \frac{-35}{7} = -5 \]

### Step 3: Write the polynomial in factor form
The polynomial can be expressed in terms of its zeroes:
\[ P(x) = (x - α)(x - β) \]
Substituting the values of α and β:
\[ P(x) = (x - 7)(x + 5) \]

### Step 4: Expand the polynomial
Now, we will expand the expression:
\[ P(x) = x^2 + 5x - 7x - 35 \]
Combining like terms:
\[ P(x) = x^2 - 2x - 35 \]

### Final Polynomial
Thus, the polynomial is:
\[ P(x) = x^2 - 2x - 35 \]