Question

If a-b, a and a+b are the zeroes of the polynomial x^3-3x^2+x+1 the value of (a+b) is

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

a-b, a and a+b are the zeroes of the polynomial x^3-3x^2+x+1

To Find:

The value of (a+b)

Solution:

polynomial f(x) = x³ - 3x² + x + 1 .

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

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

As we know,

α + β + γ = -b/a .

⇒ ( a - b ) + a + ( a - b ) = -(-3)/1 .

⇒ 3a = 3 .

⇒ a = 3/3 .

∴ a = 1 .

And,

αβ + βγ + γα = c/a .

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

⇒ a² - ab + a² + ab + a² - b² = 1 .

⇒ 3a² - b² = 1 .

⇒ ( 3 × 1² ) - b² = 1 .         { ∵ a = 1 }

⇒ 3 - b² = 1 .

⇒ b² = 3 - 1 .

⇒ b² = 2 .

∴ b = ±√2 .

As a = 1 and b = ±√2 .

( a + b ) = ( 1 ±√2 )

Answer 2

$\huge{\fbox{\fbox{\red{\star\;Answer}}}}$

• If ax³+bx²+cx+d is a cubic polynomial then sum of roots = -b/a

a-b+a+a+b = 3

3a = 3

a = 1

• Product of roots = -d/a

(a-b)a(a+b) = -1

1-b² = -1

b² = 2

b = + or -√2

• The required value a + b = 1 + √2 or 1 - √2

Hope it helps

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$\huge{\fbox{\fbox{\blue{\star\; Ravan\;Maharaj}}}}$