Question

If the zeros of the polynomial x³-3x²+x+1 are a-b,a,a+b . Find a and b !​

Answer 1

Answer:

Step-by-step explanation:

Given polynomial :-

f(x) = x³ - 3x² + x + 1

Solution :-

Here  a = 1 , b = -3 , c = 1 , d = 1 .

Let α = (a - b) , β = a and γ = (a + b)

We know that,

⇒ α + β + γ = -b/a .

⇒ (a - b) + a + (a - b) = -(-3)/1 .

⇒ 3a = 3 .

⇒ a = 3/3 .

⇒ a = 1

Also we know that,

⇒ αβ + βγ + γα = c/a .

⇒ a(  - b) + a(a + b)  + (a + b)(a - b) = 1/1 .

⇒ a² - ab + a² + ab + a² - b² = 1 .

⇒ 3a² - b² = 1 .

Putting a = 1, we get

⇒ (3 × 1²) - b² = 1 .        

⇒ 3 - b² = 1 .

⇒ b² = 3 - 1 .

⇒ b² = 2 .

⇒ b = ±√2 .

Hence, a = 1 and b = ±√2.

Answer 2

AnswEr :

For any cubic polynomial in the form of :

⋆ p( x ) = px³ + qx² + rx + s

with α, β and γ as zeros of polynomial.

• α + β + γ = - q /p
• αβ + βγ + γα = r /p
• αβγ = - s /p

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• Let's Head to the Question Now :

⋆ p( x ) = x³ - 3x² + x + 1

where, (a - b), a, (a + b) are zeros.

And the Terms are :

• p = 1
• q = - 3
• r = 1
• s = 1

• By Addition of zeros we get :

⇒ α + β + γ = - q /p

⇒ (a - b) + a + (a + b) = - (- 3) /1

⇒ 3a = 3

⇒ a = 3 /3

⇒ a = 1

• By Multiplication of zeros we get :

⇒ αβγ = - s /p

⇒ (a - b) × a × (a + b) = - 1 /1

• plugging the values of a

⇒ (1 - b)(1 + b) × 1 = - 1

• (a - b)(a + b) = (a² - b²)

⇒ ( 1 )² - ( b )² = - 1

⇒ 1 - b² = - 1

⇒ 1 + 1 = b²

⇒ b² = 2

⇒ b = ±√2

∴ Value of a is 1 and, value of b is ±√2.