Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#4
i. Find the quotient when is divided by , and show that the remainder is 5.
ii. Show that the equation has exactly one real root.
Hence quotient is and remainder is .
We are required to show that following equation has exactly one real roots;
When a polynomial, , is divided by a non-constant divisor, , the quotient and the remainder are defined by the identity;
As demonstrated in (i), we can see that;
It is evident that if we divide by and remainder is ZERO then must be smaller by remainder .
Now equation on left hand side is same as the given equation.
Now we have two options.
We further need to factorise the quadratic factor.
For a quadratic equation , the expression for solution is;
Where is called discriminant.
If , the equation will have two distinct roots.
If , the equation will have two identical/repeated roots.
If , the equation will have no roots.
Therefore, we find discriminant of this quadratic equation.
Hence, this quadratic factor of given equation will have no real roots.
From the other option , we can see that;