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5 and –3 are the zeroes of a quadratic polynomial p(x). Can we form the quadratic polynomial with the help of these zeroes?
Yes, we can. Let us see how we can form the polynomial.
5 is a zero of the polynomial, it means that (x – 5) is a factor of the polynomial p(x). Similarly (x + 3) is also a factor of the polynomial p(x). We know that a quadratic polynomial can have only two linear factors. Thus, we can write the polynomial p(x) as follows:
p(x) = (x – 5) (x + 3)
= x2 + (– 5 + 3) x + (– 5) (3)
[Using the identity (x + a) (x + b) = x2 + (a + b) x + ab, where a = – 5 and b = 3]
Thus, p(x) = x2 – 2x – 15
This is the required quadratic polynomial.
If we know the zeroes of the quadratic polynomial, then we can find the polynomial as above. On the other hand, if we know the sum and product of the zeroes of the polynomial, then we can use the following formula to find the polynomial.
p(x) = x2 – (sum of zeroes)x + product of zeroes
For example: if the sum and the product of the zeroes of a polynomial are 2 and –8 respectively, then find the polynomial.