Question

If α and β are the zeroes of the quadratic polynomial f(x) = x2 – x – 2, find a polynomial whose

zeroes are 2α + 1 and 2β + 1.​

Answer 1

Step-by-step explanation:

EXPLANATION.

α, β are the zeroes of the quadratic polynomial.

⇒ f(x) = x² - x - 2.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = -(-1)/1 = 1. - - - - - (1).

Products of the zeroes of the quadratic equation.

⇒ αβ = c/a.

⇒ αβ = (-2)/1 = - 2. - - - - - (2).

To find polynomial whose zeroes are,

⇒ 2α + 1  and  2β + 1.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ α + β = -b/a.

⇒ 2α + 1 + 2β + 1.

⇒ 2α + 2β + 2.

⇒ 2(α + β) + 2.

Put the value of α + β = 1 in the equation, we get.

⇒ 2(1) + 2 = 4.

⇒ α + β = 4. - - - - - (3).

Products of the zeroes of the quadratic equation.

⇒ αβ = c/a.

⇒ (2α + 1)(2β + 1).

⇒ (2α)(2β) + (2α)(1) + (1)(2β) + (1)(1).

⇒ 4αβ + 2α + 2β + 1.

⇒ 4(αβ) + 2(α + β) + 1.

Put the value of αβ = - 2 and α + β = 1 in the equation, we get.

⇒ 4(-2) + 2(1) + 1.

⇒ - 8 + 2 + 1.

⇒ - 8 + 3 = - 5.

⇒ αβ = - 5. - - - - - (4).

As we know that,

Formula of the quadratic polynomial.

⇒ x² - (α + β)x + αβ.

Put the value in the equation, we get.

⇒ x² - (4)x + (-5) = 0.

⇒ x² - 4x - 5 = 0.

Answer 2

Answer:

alpha + beeta = -b/a

= - (-1)/1

= 1 equation 1

alpha × beeta = c/a

= - 2 / 1

= - 2 equation 2

2alpha +1 and 2 beeta + 1

sum of these zeroes = 2alpha+1+2beeta+1

=2+ 2alpha + 2 beeta

= 2+ 2(alpha + beeta )

=2+2( 1 ) from equation 1

=2+2

=4

Product of these zeroes= (2alpha +1)(2beeta+1)

= 4 alpha. beeta + 2 alpha + 2 beeta + 1

= 4( -2 ) + 2( alpha + beeta ) +1 from eqn 2

= -8+ 2(1) +1

= - 8 +2 + 1

= - 5

quadratic polynomial= x² + (sum of zeroes )x + product of zeroes

= x² + ( 4 ) x + (- 5 )

= x² + 4x - 5

hope it helps you