If alpha and beta are the zeroes of the quadratic polynomial :p(x) =3x^2 - 4x 1, find a quadratic polynomial whose zeroes are alpha^2/beta and beta^2/alpha ?

Question

Answer

P(x)=3x^2-4x+1,
by splitting the middle term,
3x^2-3x-x+1=0,
3x(x-1)-1(x-1)=0,,
(3x-1)(x-1)=0,
3x-1=0, or, x-1=0,
x=1/3, or, x=1,
or the zeros of the quadratic polynomial p(x)= 3x^2-4x+1 are 1/3 and 1,
let the new polynomial be f(x) having zeroes as ,
(1/3)^2/1 and 1^2/1/3,
1/9 and 3,
f(x)=x^2-( sum of zeroes)x+ product of zeroes,
=x^2-(1/9+3)x+1/9×3,
=x^2-28/9x+3/9,
=9x^2-28x+3

Answer

Answer

For the quadratic equation with roots, α ,β
Sum of roots = 4/3
Product of roots = 1/3 .

Now, The quadratic equation with roots α²/β ,
β²/ α

Now,
Sum of roots = α²/β + β²/ α = α³+β³/αβ
= (α + β)³-3αβ(α+β)/αβ
= (4/3)³-3(1/3)(4/3) / 1/3
= 64/27 - 4/3 / 1/3
= 64/27-36/27 / 1/3
= 28/27 * 3/1
= 28/9

Product of roots = (α²/β * β²/ α)
= (αβ )
= 1/3 .

The quadratic equation with required roots
= x²-28/9x + 1/3
= 9(x² - 28/8x+1/3)
= 9x² - 28x + 3

That's all🙂

Answer

Answer

Α and β are the zeros of 3x²-4x +1 polynomial,

first of all we factorise 3x²-4x+1
3x² -4x + 1

=3x² -3x -x +1

=3x( x -1) -1(x -1)

=(3x -1)(x -1)

hence. (3x -1) and (x -1) are the factors of given polynomial .

so, x = 1/3 and 1 are the zeros of that polynomial.

hence, α = 1/3. and β = 1
or α = 1 and β. = 1/3
you can choose any one in both

I choose α = 1. and β = 1/3

now,
let any unknown. polynomial. whose zeros are α²/β and β²/α

α²/β = (1)²/(1/3) = 3

β²/α = (1/3)²/1 = 1/9

now, equation of unknown polynomial.

x²- ( sum of roots)x + product of roots

= x²- ( α²/β + β²/α)x +(α²/β)(β²/α)

put α²/β = 3 and β²/α = 1/9

= x²- ( 3 +1/9)x + 3 × 1/9

= x² -28x/9 + 3/9

={ 9x² -28x + 3 }1/9

hence, 9x² -28x + 3 is answer

I hope it's helpful for us....

Answer

Explanation:
To find a quadratic polynomial whose zeroes are alpha^2/beta and beta^2/alpha, we can use the sum and product of roots formula.

Sum of roots formula: alpha + beta = -b/a

Product of roots formula: alpha * beta = c/a

We can use these formulas to find expressions for alpha^2/beta and beta^2/alpha.

Expression for alpha^2/beta:
alpha + beta = -(-4)/3 = 4/3
alpha * beta = 1/3
alpha^2/beta = alpha * alpha/beta = (alpha/beta) * alpha = (1/3)/(4/3) * alpha = alpha/4

Expression for beta^2/alpha:
alpha + beta = 4/3
alpha * beta = 1/3
beta^2/alpha = beta * beta/alpha = (beta/alpha) * beta = (1/3)/(4/3) * beta = beta/4

Now, we can use the sum and product of roots formula again to find the quadratic polynomial.

Sum of roots formula: alpha^2/beta + beta^2/alpha = (-b/a) * (1/a)
4/alpha = (-b/a) * (1/a)
-b = 4

Product of roots formula: alpha^2/beta * beta^2/alpha = c/a
(alpha * beta)^2 = 1/3 * 1/3
c = 1/9

Therefore, the quadratic polynomial whose zeroes are alpha^2/beta and beta^2/alpha is:
p(x) = 3x^2 + 4x/alpha + 1/9

Final Answer:
The quadratic polynomial whose zeroes are alpha^2/beta and beta^2/alpha is p(x) = 3x^2 + 4x/alpha + 1/9.

Answer

Α and β are the zeros of 3x^2-4x +1 polynomial,

first of all we factorise 3x^2-4x+1

3x^2 -4x + 1

=3x^2 -3x -x +1

=3x( x -1) -1(x -1)

=(3x -1)(x -1)

hence. (3x -1) and (x -1) are the factors of given polynomial .

so, x = 1/3 and 1 are the zeros of that polynomial.

hence, α = 1/3. and β = 1

or α = 1 and β. = 1/3

you can choose any one in both

I choose α = 1. and β = 1/3

now,

let any unknown. polynomial. whose zeros are α^2/β and β^2/α

α^2/β = (1)^2/(1/3) = 3

β^2/α = (1/3)^2/1 = 1/9

now, equation of unknown polynomial.

x^2- ( sum of roots)x + product of roots

= x^2- ( α^2/β + β^2/α)x +(α^2/β)(β^2/α)

put α^2/β = 3 and β^2/α = 1/9

= x^2- ( 3 +1/9)x + 3 ^2 1/9

= x^2 -28x/9 + 3/9

={ 9x^2 -28x + 3 }1/9

hence, 9x^2 -28x + 3 is answer

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