Question

If the zeroes of the polynomial x³-3x²+x+1 are (a-b),a and (a+b), then find the values of a and b.

Answer 1

Answer :-

The value of a is 1 and the value of b is √2 or - √2

Explanation :-

x³ - 3x² + x + 1

Zeroes of the polynomial are a, (a - b) and (a + b)

Comparing x³ - 3x² + x + 1 with ax³ + bx² + cx + d

We get

• a = 1

• b = - 3

• c = 1

• d = 1

We know that

Sum of the zeroes = - b/a

⇒ a + (a - b) + (a + b) = - (-3)/1

⇒ a + a - b + a + b = 3

⇒ 3a = 3

⇒ a = 3/3

⇒ a = 1

We know that

Product of zeroes = - d/a

⇒ a(a - b)(a + b) = - 1/1

⇒ a(a² - b²) = - 1 [Since (a + b)(a - b) = a² - b²]

⇒ 1(1² - b²) = - 1 [Since a = 1]

⇒ 1² - b² = - 1

⇒ 1 - b² = - 1

⇒ 1 + 1 = b²

⇒ 2 = b²

⇒ ± √2 = b

⇒ b = ± √2

Therefore the value of a is 1 and the value of b is √2 or - √2

Answer 2

$\large{\mathfrak{\underline{\underline{\blue{Answer:-}}}}}$

a = 1

b = √2

${\mathfrak{\underline{\underline{\pink{Step-By-Step-Explanation:-}}}}}$

$\large{\mathtt{\green{Given:-}}}$

x³ - 3x² + x + 1 = 0

Zeroes are :- (a - b) , a, (a + b)

____________________________

$\large{\mathtt{\green{To \: find:-}}}$

Values of a and b

___________________________

$\large{\mathtt{\green{Proof:-}}}$

We know that

x³ - 3x² + x + 1 = 0

And,

a = 1

b = -3

c = 1

d = 1

_________________________

➭ Sum of zeroes = -b/a

________[Put Values]

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

⇒ 3a = 3

⇒ a = 3/3

⇒ a = 1

$\large{\star{\underline{\boxed{\red{a \: = \: 1}}}}}$

___________________________

➭ Product of zeroes = -d/a

_________[Put Values]

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

⇒ (a² - b²) * a = -1

____________[Put value of a]

⇒ (1² - b²) * 1 = -1

⇒ 1 - b² = -1

⇒ -b² = -1-1

⇒ -b² = -2

⇒ b² = 2

⇒ b = √2

$\large{\star{\underline{\boxed{\red{b \: = \: {\sqrt{2}}}}}}}$

___________________________