Question

Find a quadratic polynomials each with the given numbers as the sum and product of its zeroes respectively.
1/4,-1
​

Answer 1

Given that,

• Sum of Quadratic polynomial = ¼

• & Product of Quadratic polynomial = – 1.

⠀

$\underline{\bigstar\:\textsf{Required Quadratic Polynomial \: :}} \\ \\ \\ :\implies\sf p(x) = x^2 - (\alpha + \beta)x + \alpha \: \beta \\\\\\:\implies\sf p(x) = x^2 - \dfrac{1}{4}x + (- 1) \\\\\\:\implies\sf p(x) = x^2 - \dfrac{1}{4}x - 1 \\\\\\:\implies\underline{\boxed{\frak{\purple{\pmb{p(x) = 4x^2 - x -4}}}}}\;\bigstar$

⠀

$\therefore{\underline{\sf{Hence,~the~Quadratic~polynomial~is~ \sf {\pmb{4x^2 - x - 4}.}}}}$

$\rule{100px}{.3ex}$

⠀

An Quadratic equation can be represent as in the form of (ax² + bx + c = 0).

If α and β are roots of any Quadratic equation (ax² + bx + c = 0) then Sum and Product is given by :

⠀⠀⠀⠀⋆ Sum (α + β) = (–b)/a

⠀⠀⠀⠀⋆ Product (α β) = c/a

Answer 2

Answer:

• 4x² - x - 4

Step-by-step explanation:

⇒ Let the required polynomial be,

• p(x) = ax² + bx + c

Now, sum of zeroes, (assume a = 1)

• -b/a
• -b/1 = 1/4
• b = -1/4

Now, product of zeroes, (assume a = 1)

• c/a
• c/a = -1
• c/1 = -1
• c = -1

Hence, we have,

• a = 1
• b = -1/4
• c = -1

Now, the required polynomial will be,

• ax² + bx + c
• (1)x² + (-1/4)x + (-1)
• x² - 1/4x - 1 (take 4 common)
• 4(x² - 1/4x - 1)
• 4x² - x - 4

∴ The quadratic polynomial is 4x² - x - 4.