Question

if alpha and beta are the zeros of quadratic polynomial f(x)= x square +px+q form a polynomial whose zeros are alpha+2 and beta+2​

Answer 1

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;

Then;

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

Product of zeros = A•B = c/a

Here, the given polynomial is;

f(x) = x^2 + px + q

It is given that,

alpha and beta are the zeros of the given polynomial f(x).

Thus,

Sum of zeros = alpha + beta = -p/1= -p

Product of zeros = alpha•beta = q/1=q

Also,

It is provided that, (alpha+2) and (beta+2) are the zeros of required polynomial.

Thus, for the required polynomial,

We have,

Sum of zeros =(alpha+2) + (beta+2)

= alpha + beta + 4

= -p + 4

= 4 - p

Also,

Product of zeros= (alpha+2)•(beta+2)

= alpha•beta+2•alpha

+ 2•beta + 4

= alpha•beta + 4

+ 2(alpha + beta)

= q + 4 + 2(-p)

= q - 2p + 4

Note:

Any quadratic polynomial given by;

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

Thus,

The required polynomial will be;

x^2 - (4 - p)x + (q - 2p + 4)

OR

x^2 + (p - 4)x + (q - 2p + 4)