Question

if alpha and beta are the zeroes of the polynomial 2x ²-5×+7,then find a quadratic polynomial whose zeroes are 3alpha +4beta and 4alpha+3beta​

Answer 1

Answer:

Let a=alpha,b=beta. from given question , sum of roots= a+b= 5/2. product of roots =ab=7/2

X²-[(3a+4b)+(4a+3b)] X + [(3a+4b)(4a+3b)]

x^2- [7(a+b)] X + [12a^2+12b^2+25ab]

x^2-(35/2) X + [12 {(a+ b)^2-2ab}+175/2

x^2-(35/2) X +[75-7]+175/2

X^2 - (35/2) X+ 311/2

2 x^2-35x+311

Answer 2

Polynomial → 2x² - 5x + 7

On factorizing the above polynomial,

2x² - 5x + 7 = 0

⇒ 2x² + 2x - 7x + 7 = 0

⇒ 2x(x + 1) - 7(x + 1) = 0

⇒ (2x - 7)(x + 1) = 0

Now,

(2x - 7) = 0          OR        (x + 1) = 0

→ 2x = 7               OR        → x = 0 - 1

→ x = 3.5               OR         → x = -1

Let,

• α = 3.5
• β = -1

For finding 3α + 4β,

3α + 4β = 3(3.5) + 4(-1)

3α + 4β = 10.5 - 4

3α + 4β = 6.5

∴ 3α + 4β = 6.5

For finding 4α + 3β,

4α + 3β = 4(3.5) + 3(-1)

4α + 3β = 14 - 3

4α + 3β = 12

∴ 4α + 3β = 12

Now we have to find a polynomial whose zeroes are 6.5 and 12

Sum of zeroes = 12 + 6.5 = 18.5

Product of zeroes = 12 × 6.5 = 78

Polynomial is of the form;

x² - (Sum of zeroes)x + Product of zeroes

→ x² - 18.5x + 78 = 0

Multiply by 2,

2x² - 37x + 156 = 0

is the required polynomial