Question

Solve the q in the photo

Answer 1

p(x) = x² + 4x + 2

Given that ɑ and β are the zeroes of p(x)

To Find,

Polynomial with zeroes 2ɑ + β and ɑ + 2β

Solution:

We know,

==> Sum of zeroes = -(coefficient of x) / (coefficient of x²)

==> ɑ + β = -4 ...(1)

Also,

==> Product of zeroes = (constant term) / (coefficient of x² )

==> ɑβ = 2 ...(2)

Squaring in eq (1) both sides,

==> ɑ² + β² + 2ɑβ = 16

==> ɑ² + β² = 16 - 4 { from (2) , ɑβ = 2 }

==> ɑ² + β² = 12 ...(3)

Now,

A quadratic polynomial is in the form of :

==> x² - (sum of zeroes)x + (product of zeroes)

==> Sum of zeroes = 2ɑ + β + a + 2β

==> sum of zeroes = 3(ɑ + β)

==> sum of zeroes = 3 × -4 { from (1) }

==> sum of zeroes = -12 ...(4)

Also,

==> Product of zeroes = (2ɑ + β) (ɑ + 2β)

==> 2ɑ² + 2ɑβ + ɑβ + 2β²

==> 2ɑ² + 2β² + 2×2 + 2 { from (2) , ɑβ = 2 }

==> 2ɑ² + 2β² + 6

==> 2(ɑ² + β²) + 6

==> 2 × 12 + 6 { from (3) }

==> 24 + 6

==> product of zeroes = 30 ...(5)

Now,

Required polynomial = x² - (sum of zeroes)x + (product of zeroes)

==> x² - (-12)x + 30

==> x² + 12x + 30

Hence, Answer to this question is x² + 12x + 30