Question

Find a quadratic polynomial each with the given numbers as the sum and product of its
zeroes respectively.
1/4,-1
ii √2,1/3
iii0,√5
iv 1,1
v -1/4,1/4
vi 4,1​

Answer 1

Step-by-step explanation:

By using formula...

x^2-Sx+p=0

1) x^2-1/4x+(-1)=0

x^2-1x-1=0*4

x^2-1x-1=0

2) x^2-√2x+1/3=0

x^2-√2x+1=0*3

x^2-√2x+1=0

3) x^2-0x+√5=0

x^2-√5

4) x^2-1x+1=0

5) x^2-(-1/4)x+1/4=0

x^2+1/4x+1/4=0

x^2+1x+1=0

6) x^2-4x+1=0

When a equation is form like ax^2+bx+c=0 you can remove that zero on right side .

Answer 2

↬ Solution :-

❍  (i) 1/4 , -1

➝ α + β = 1/4

➝ α × β = -1

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - 1/4 x + (-1)

➝ x² - x/4 - 1

➝ $\rm\:\dfrac{{4x}^{2}-x-4}{4}\:\:\:\:\:\:\:[multiplying\:by\:4]$

➝ 4x² - x - 4

Hence, required polynomial will be 4x² - x -4.

❍  (ii) √2 , 1/3

➝ α + β = √2

➝ α × β = 1/3

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - √2 x + 1/3

➝ x² - √2x +1/3

➝ $\rm\dfrac{3{x}^{2}-3\sqrt{2x}+1}{3}\:\:\:\:\:[Multiplying\:by\:3]$

➝ 3x² - 3√2x +1

Hence, required polynomial will be 3x² - 3√2x + 1.

❍ (iii) 0 , √5

➝ α + β = 0

➝ α × β = √5

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - 0 x + 5

➝ x² + √5

Hence, required polynomial will be x² + √5

❍ (iv) 1 , 1

➝ α + β = 1

➝ α × β = 1

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - 1 x + 1

➝ x² - x + 1

Hence, required polynomial will be x² - x + 1.

❍ (v) -1/4 , 1/4

➝ α + β = -1/4

➝ α × β = 1/4

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - (-1/4)x + 1/4

➝ x² + x/4 + 1/4

➝ $\rm\:\dfrac{{4x}^{2}+x+1}{4}\:\:\:\:\:[multiplying\:by\:4]$

➝ 4x² + x + 1

Hence, required polynomial will be 4x² + x + 1.

❍ (vi) 4 , 1

➝ α + β = 4

➝ α × β = 1

⸕ In Quadratic polynomial,

We know that,

quadratic polynomial = x² - ( α + β )x + α × β

➝ x² - 4 . x + 1

➝ x² - 4x + 1

Hence, required polynomial will be x² - 4x +1.