Question

Question :-

1) :- If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1 the remainder comes out to be ( ax + b) then find the value of a and b.


2) :- If α and β are the zeroes of the quadratic polynomial f(x) = x² + x - 2, then find a polynomial whose zeroes are 2α + 1 and 2β + 1.
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Answer 1

Step-by-step explanation :-

Question :-

1) :- If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1 the remainder comes out to be ( ax + b) then find the value of a and b.

Answer :-

→ See attachment 1.

→ a = 1 and b = 2.

Question :-

2) :- If α and β are the zeroes of the quadratic polynomial f(x) = x² + x - 2, then find a polynomial whose zeroes are 2α + 1 and 2β + 1.

Answer :-

→ See attachment 2.

→ x² - 9.

Hence, it is solved.

Answer 2

Step-by-step explanation:

1) If the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 is divided by another polynomial 3x² + 4x + 1 the remainder comes out to be ( ax + b) then find the value of a and b.

⇒ Let p(x) be 6x⁴ + 8x³ + 17x² + 21x + 7 and g(x) is 3x² + 4x + 1

3x² + 4x + 1 ) 6x^⁴ + 8x^³ + 17x^² + 21x + 7 (2x^² + 5

6x^⁴ + 8x^² + 2x^²

- - -

---------------

15x^² + 21x + 7

15x^² + 20 x + 5

- - -

---------------

x + 2

⇒ The remainder is x + 2

According to the question given the remainder is (ax + b)

On comparing both the remainders

We get,

a as 1 and b as 2

a = 1 & b = 2

Therefore, the value of a is 1 and b is 2 respectively.

2) If α and β are the zeroes of the quadratic polynomial f(x) = x² + x - 2, then find a polynomial whose zeroes are 2α + 1 and 2β + 1.

⇒ Let f(x) be x^² + x - 2

And let the two zeroes of polynomial be α and β respectively.

f(x) = x^² + x - 2 = 0

= x^² + 2x - x - 2 = 0

= x(x + 2) - 1(x - 2) = 0

= (x -1) (x + 2)

x = 1 or x = - 2

α = 1 and β = - 2 respectively.

According to Find the question polynomial whose zeroes are 2α + 1 and 2β + 1

2α + 1

2(1) + 1

2 + 1

3

2β + 1

2( -2 ) + 1

- 4 + 1

- 3

New polynomial using the following detail

=>x^² - (sum of zeroes )x + (product of zeroes )

= x^² - (3+ (-3)x + 3(-3)

= x^² - 0x - 9

= x^² - 9

Therefore, the new polynomial formed is x^2 - 9 whose zeroes are 3 & - 3 respectively.