Question

If A and B are the zeroes of the quadratic polynomial x²- 2x + 3, find a polynomial whose zeroes are a + 2, B + 2.

A:- Alpha
B:- Beta

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Answer 1

Step-by-step explanation:

a and b are the zeros of

x

2

−2x+3

Then,

We know that

Sum of zeros =−

coeff. of x

2

coeff. of x

a+b=−

+1

−2

a+b=2−−−−−−−−(1)

Now,

Product of zeros =

coeff. of x

2

constant term

a.b=

1

3

ab=3−−−−−−−−−−−(2)

If 2a+3 and 2b+3 are the zeros of other polynomial.

Then

Sum of zeros =2a+3+2b+3

=2(a+b)+6

=2(2)+6

=10

Sum of zeros =10

Product of zeros =(2a+3)(2b+3)

=4ab+6a+6b+9

=4ab+6(a+b)+9

=4×3+6×2+9

=12+12+9

=24+9

Product of zeros =33

Now,

Equation of polynomial

x

2

− (Sum of zeros) x+ product of zeros =0

x

2

−10x+33=0

Answer 2

Answer:

x² - 7x + 13

Step-by-step explanation:

Given,

          α, β are zeroes of x² - 3x + 3

Comparing x² - 3x + 3 with ax² + bx + c,

                                         a = 1, b = -3, a = 3

We know that sum of zeroes = α + β = -b/a = -(-3)/1 = 3

And product of zeroes = αβ = c/a = 3/1 = 3

Now,

       Polynomial with zeroes α + 2, β + 2:

       = [ x - (α + 2) ] [ x - (β + 2) ]

       = x² - x (β + 2)  - x (α + 2) + ( α + 2 ) ( β + 2)

       = x² - x ( α + 2 + β + 2 ) + ( α + 2 ) ( β + 2 )

       = x² - x ( α + β + 4 ) + αβ + 2α + 2β + 4

       = x² - x ( 3 + 4 ) + 3 + 2 ( α + β ) + 4

       = x² - 7x + 3 + 2 ( 3 ) + 4

       = x² - 7x + 3 + 6 + 4

       = x² - 7x + 13

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