To determine which expression is a polynomial with given roots, we can use the fact that if a polynomial has roots, then the polynomial can be formed by multiplying factors corresponding to those roots.
In this case, we have the roots as √3, √3, and 2. This means the polynomial can be represented as:
- (x – √3)(x – √3)
- (x – 2)
Now, we can multiply these factors to express the polynomial in standard form. First, we handle the double root:
(x - √3)(x - √3) = (x - √3)² = x² - 2√3x + 3
Next, we can multiply this result by the factor for the root at 2:
(x² - 2√3x + 3)(x - 2)
Now, we distribute:
= x³ - 2x² - 2√3x² + 4√3x + 3x - 6
Combining like terms, we arrive at:
= x³ - (2 + 2√3)x² + (4√3 + 3)x - 6
This final expression is a polynomial representation with the roots √3, √3, and 2. If given multiple choices, look for a polynomial similar in structure to this derived expression.