To determine a polynomial with the specified roots, we start by recalling that if a number is a root of a polynomial, then the polynomial can be expressed as a product involving that root. In this case, the roots given are
√5 (repeated root) and 3. The general form for a polynomial derived from its roots can be expressed as:
P(x) = (x – r1)(x – r2)(x – r3)…
Here, r1, r2, and r3 are the roots of the polynomial. For our case:
- r1 = √5
- r2 = √5
- r3 = 3
This means we can express the polynomial as:
P(x) = (x – √5)(x – √5)(x – 3)
Since the root √5 is repeated, it is crucial to square the factor corresponding to it:
P(x) = (x – √5)²(x – 3)
Next, we can expand this polynomial:
- First, calculate (x – √5)²:
- (x – √5)(x – √5) = x² – 2√5x + 5
Now, we will multiply this result by (x – 3):
P(x) = (x² – 2√5x + 5)(x – 3)
Expanding this expression:
- x² * (x – 3) = x³ – 3x²
- -2√5x * (x – 3) = -2√5x² + 6√5x
- 5 * (x – 3) = 5x – 15
Putting it all together:
P(x) = x³ – 3x² – 2√5x² + 6√5x + 5x – 15
Now, combine like terms:
P(x) = x³ + (-3 – 2√5)x² + (6√5 + 5)x – 15
Therefore, the polynomial that has roots √5, √5, and 3 is:
P(x) = x³ + (-3 – 2√5)x² + (6√5 + 5)x – 15.