# 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

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**Question**

** 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.

**Solution**

** i.
**

Hence quotient is and remainder is .

** ii.
**

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 .

Therefore;

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;