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Express your expression as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 2

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Express your expression as a single fraction in its simplest form.

Worked Solution & Example Answer:Express your expression as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2010 - Paper 2

Step 1

Combine the two fractions

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Answer

To combine the fractions, we need a common denominator. The common denominator for the two fractions ( \frac{x+1}{3x^2 - 3} ) and ( \frac{1}{3x+1} ) is ( (3x^2 - 3)(3x + 1) ). Thus, we rewrite the expression:

x+13(x2−1)(3x+1)−1(3x2−3)(3x2−3)(3x+1)\frac{x + 1}{3(x^2 - 1)(3x + 1)} - \frac{1(3x^2 - 3)}{(3x^2 - 3)(3x + 1)}

Step 2

Simplify the fractions

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Answer

Notice that ( 3(x^2 - 1) ) factors to ( 3(x - 1)(x + 1) ). This means that:

x+1−(3x2−3)3(x2−1)(3x+1)\frac{x + 1 - (3x^2 - 3)}{3(x^2 - 1)(3x + 1)}

Substituting in gives:

x+1−3x2+33(x−1)(x+1)(3x+1)=−3x2+x+43(x−1)(x+1)(3x+1)\frac{x + 1 - 3x^2 + 3}{3(x - 1)(x + 1)(3x + 1)} = \frac{-3x^2 + x + 4}{3(x - 1)(x + 1)(3x + 1)}

Step 3

Final Simplification

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Answer

The final step is to ensure the numerator is expressed correctly:

−3x2+x+43(x−1)(x+1)(3x+1)\frac{-3x^2 + x + 4}{3(x - 1)(x + 1)(3x + 1)}

Thus, our expression can be written as:

4−3x23(x−1)(x+1)(3x+1)\frac{4 - 3x^2}{3(x - 1)(x + 1)(3x + 1)}

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