Answer:
- A. P(x) = x⁵ + 3x⁴ - 13x³ - 39x² + 40x + 120
Step-by-step explanation:
With the given zero's, the polynomial is:
- P(x) = (x - 2√2)(x - √5)(x + 3)
Open parenthesis in steps:
1. (x - 2√2)(x - √5) = x² - (2√2 + √5)x + 2√10 = x² - 2x√2 - x√5 + 2√10
2. (x² - 2x√2 - x√5 + 2√10)(x + 3) =
x³ - 2√2x² - √5x² + 2√10x + 3x² - 6√2x - 3√5x + 6√10 =
x³ + 3x² - 2x²√2 - x²√5 - 6x√2 - 3x√5 + 2x√10 + 6√10
This is not matching any options so we add two more zero's: -2√2 and -√5
New polynomial looks like:
- P(x) = (x - 2√2) (x + 2√2)( x - √5)(x - √5)(x + 3)
Open parenthesis in steps:
- 1. (x - 2√2) (x + 2√2) = x² - 8
- 2. (x - √5) (x + √5) = x² - 5
- 3. (x² - 8)(x² - 5) = x⁴ - 8x² - 5x² + 40 = x⁴ - 13x² + 40
- 4. (x⁴ - 13x² + 40)(x + 3) = x⁵ + 3x⁴ - 13x³ - 39x² + 40x + 120
- P(x) = x⁵ + 3x⁴ - 13x³ - 39x² + 40x + 120
