Question

Find a quadratic polynomial whise zeros are 3 + root 2 and 3 - root 2

Answer 1

Step-by-step explanation:

Form of a quadratic polynomial

x² - (Sum of roots)x + (multiple of roots) = 0

x² - (3+√2 + 3-√2)x + (3+√2)(3-√2) = 0

x² - 6x + 7 = 0

Answer 2

Question: Find a quadratic polynomial whose zeros are (3 + √2) and (3 - √2).

Solution:

The quadratic polynomial can be found out by using the general formula  to frame a quadratic equation when the zeroes are given.

General Formula:  (x² - (α + β)x + αβ)

Let us take α and β to be the zeroes. Therefore, let 3 + √2 be 'α' and 3 - √2 be 'β'.

Sum of the zeroes = α + β

⇒ α + β = (3 + √2) + (3 - √2)

⇒ α + β = 3 + √2 + 3 - √2

⇒ α + β = 3 + 3

⇒ α + β = 6

Product of the zeroes = α × β

⇒ α × β = (3 + √2) + (3 - √2)

Using the formula (a + b) (a - b) = a² - b²

⇒ α × β = (3)² - (√2)²

⇒ α × β = 9 - 2

⇒ α × β = 7

Using the general formula we get,

$\sf Quadratic \ Polynomial \Longrightarrow \ k\left(x^2 - (\alpha+\beta)x + \alpha \beta \right)\\ \\\sf Quadratic \ Polynomial \Longrightarrow \ k\left(x^2 - (6)x + 7 \right)\\ \\\sf Quadratic \ Polynomial \Longrightarrow \ k\left(x^2 - 6x + 7 \right)\\ \\ \sf If \ k = 1,\\ \\\underline{\underline{\sf Quadratic \ Polynomial \Longrightarrow \ x^2 - 6x + 7}}$

Final Answer:  x² - 6x + 7.