Find the polynomial whose sum and product of roots are 6 and -16?

Options

A. `x^(2)+6x`
B. `x^(2)-6x-16`
C. `x^(2)-6x+16`
D. `x^(2)-10x+96`

Answer

To find the polynomial whose sum and product of roots are given as 6 and -16, we can follow these steps:

### Step 1: Define the roots
Let the roots of the polynomial be α (alpha) and β (beta). According to the problem, we know:
- Sum of roots (α + β) = 6
- Product of roots (α * β) = -16

### Step 2: Write the polynomial in terms of its roots
The polynomial can be expressed using its roots as:
\[ P(x) = (x - α)(x - β) \]

### Step 3: Expand the polynomial
Using the identity for the product of two binomials, we can expand the polynomial:
\[ P(x) = x^2 - (α + β)x + (α * β) \]

### Step 4: Substitute the values of sum and product of roots
Now, substitute the known values of the sum and product of the roots into the polynomial:
- Substitute α + β = 6
- Substitute α * β = -16

So, we have:
\[ P(x) = x^2 - (6)x + (-16) \]

### Step 5: Simplify the polynomial
Now, simplify the expression:
\[ P(x) = x^2 - 6x - 16 \]

### Final Answer
Thus, the polynomial whose sum and product of roots are 6 and -16 is:
\[ P(x) = x^2 - 6x - 16 \]

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