# Past Papers’ Solutions | Assessment & Qualification Alliance (AQA) | AS & A level | Mathematics 6360 | Pure Core 1 (6360-MPC1) | Year 2007 | June | Q#6

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**Question**

**a. **The polynomial f(x) is given by .

** i. **Use the Factor Theorem to show that (x-1) is a factor of f(x).

** ii. **Express f(x) in the form , where p and q are integers.

** iii. **Hence show that the equation f(x)=0 has exactly one real root and state its value.

**b. **The curve with equation is sketched below.

The curve cuts the x-axis at the point A (1, 0) and the point B (2, 11) lies on the curve.

** i. **Find

** ii. **Hence find the area of the shaded region bounded by the curve and the line AB .

**Solution**

**a.
**

** i.
**

Factor theorem states that if is a factor of then;

For the given case is factor of .

Here and . Hence;

Hence, is factor of .

** ii.
**

If is a polynomial of degree then will have exactly factors, some of which may repeat.

We are given a polynomial of degree ,

Therefore, it will have 03 factors some of which may repeat.

From (a:i) we already have 01 factor of .

We may divide the given polynomial with any of the already known linear factor(s) to get a quadratic factor and then factorize the obtained quadratic factor to find the 2^{nd} and 3^{rd} linear factors.

For the given case;

Already known factor from (a:i) is .

We may divide the given polynomial by this factor .

Therefore, we get the quadratic factor for given polynomial.

Hence, can be written as product of linear and quadratic factors as follows;

** iii.
**

From (a:ii), we know that can be written as;

It is evident from this and (a:i) that (x-1) is a factor of f(x).

Therefore, one of the roots of f(x) can be found from;

Remaining roots of f(x) can be found from;

For a quadratic equation , the expression for solution is;

Where is called discriminant.

If , the equation will have two roots.

If , the equation will have two identical/repeated roots.

If , the equation will have no roots.

For the given equation;

Since , the equation will have no real roots.

Hence, f(x) has only one real root.

**b.
**

** i.
**

Rule for integration of is:

Rule for integration of is:

Rule for integration of is:

** ii.
**

It is evident from the diagram that;

To find the area of region under the curve , we need to integrate the curve from point to along x-axis.

We are given equation of the curve; however, we do not have equation of the line AB.

Let’s find equation of the line AB. We are given coordinates of two points on the line and .

Two-Point form of the equation of the line is;

Therefore;

Therefore;

Rule for integration of is:

Rule for integration of is:

Rule for integration of is:

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