Question

if alpha and beta are the zeros 3x^2-7x+4,find quadratic polynomial whose zeroes are (2alpha+1) and (2beta+1)​

Answer 1

Answer:

Note:

Let's consider a quadratic polynomial in variable x ,say;

ax^2 + bx + c.

If A and B are the zeros of the polynomial, then;

Sum of zeros = A+B = -b/a

Product of zeros = A•B = c/a

Here, The given polynomial is;

3x^2 - 7x + 4.

{ note: considering alpha as A and beta as B }

Thus, We have;

Sum of zeros = A + B = -(-7)/3 = 7/3

Product of zeros = A•B = 4/3

Note:

Any quadratic polynomial given by;

x^2 -(sum of zeros)x + product of zeros

Here,

It is given that;

(2A+1) and (2B+1) are the zeros of the required polynomial.

Thus, for the required polynomial

We have;

Sum of zeros = (2A+1) + (2B+1)

= 2A + 2B + 2

= 2(A+B+1)

= 2(7/3 + 1)

= 2•(10/3)

= 20/3

Product of zeros = (2A+1)•(2B+1)

= 4AB + 2A + 2B + 1

= 4AB + 2(A + B) + 1

= 4(4/3) + 2(7/3)+ 1

= 16/3 + 14/3 + 1

= 33/3

= 11

Thus, the required polynomial is;

x^2 - (20/3)x + 11

OR

(1/3)(3x^2 - 20x + 33)