Math, asked by rahul2934, 1 year ago

Given that the zeroes of the cubic polynomial x3 – 6x2 + 3x + 10 are of the form a,a + b, a + 2b for some real numbers a and b, find the values of a and b as well asthe zeroes of the given polynomial.

Answers

Answered by aquialaska
592

Answer:

Value of a = 5 & b = -3 or a = -1 & b = 3  and Zeroes are -1 , 2 , 5.

Step-by-step explanation:

Given: x³ - 6x² + 3x + 10

          Zeroes are of the form = a , a + b  , a + 2b

To find: Value of a & b and all zeroes of the polynomial.

Using relationship of coefficient and Zeroes of the polynomial we get,

Sum of the roots = -coefficient of x²/ coefficient of x³

⇒ a + 2b + a + a + b = -(-6)/1                      

⇒ 3a + 3b = 6

⇒ 3 ( a + b ) = 6

⇒ a + b = 2 ..................(1)

Product of roots = -constant/coefficient of x³

⇒ ( a + 2b )( a + b ) a = -10/1

⇒ ( a + b + b )( a + b ) a = -10

⇒ ( 2 + b )( 2 ) a = -10   ( From eqaution (1) )

⇒ ( 2 + b ) 2a = -10

⇒ ( 2 + 2 - a ) 2a = -10

⇒ ( 4 - a ) 2a = -10

⇒ 4a - a² = -5

⇒ a² - 4a - 5 = 0

Now solving obtained quadratic equation we get,

a = 5 , -1                 

Now, Putting value of a in equation in Eqn (1)

We get,

When

a = 5   ⇒  5 + b = 2 ⇒ b = -3

a = -1   ⇒  -1 + b = 2 ⇒ b = 3

when a = 5 and b = -3,  Zeroes are 5 , ( 5 + (-3) ) = 2 , ( 5 + 2(-3) ) = -1

when a = -1 and b = 3 , Zeroes are -1 , ( -1 + 3 ) = 2 , ( -1 + 2(3) ) = 5

Therefore, Value of a = 5 & b = -3 or a = -1 & b = 3  and Zeroes are -1 , 2 , 5.

Answered by smartboy3392
137

Step-by-step explanation:

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