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1. (a) Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \), (b) Hence, or otherwise, express \( \frac{3x^3 - x - 2}{x^2 - 1} \) as \( \frac{1}{x(x+1)} \) as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 4
Question 3
1. (a) Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \), (b) Hence, or otherwise, express \( \frac{3x^3 - x - 2}{x^2 - 1} \) as \( \frac{1}{x(x+1)} \) as a single fracti... show full transcript
Worked Solution & Example Answer:1. (a) Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \), (b) Hence, or otherwise, express \( \frac{3x^3 - x - 2}{x^2 - 1} \) as \( \frac{1}{x(x+1)} \) as a single fraction in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2006 - Paper 4
Step 1
Simplify \( \frac{3x^3 - x - 2}{x^2 - 1} \)
Answer
To simplify the expression, we start by factoring both the numerator and the denominator.
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Factor the denominator:
( x^2 - 1 = (x - 1)(x + 1) ) -
Factor the numerator:
We can try polynomial long division or synthetic division, but let’s look for roots using substitution. Testing values, we find that ( x = 1 ) is a root: [ 3(1)^3 - 1 - 2 = 0 ] Now we can factor the numerator as: ( 3x^3 - x - 2 = (x - 1)(3x^2 + 3x + 2) )
Thus, we have ( \frac{3x^3 - x - 2}{x^2 - 1} = \frac{(x - 1)(3x^2 + 3x + 2)}{(x - 1)(x + 1)} ).
- Cancel the common factors:
If ( x \neq 1 ), we simplify to ( \frac{3x^2 + 3x + 2}{x + 1} ).
Step 2
Hence, or otherwise, express \( \frac{3x^3 - x - 2}{x^2 - 1} \) as \( \frac{1}{x(x+1)} \)
Answer
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We already simplified ( \frac{3x^3 - x - 2}{x^2 - 1} ) to ( \frac{3x^2 + 3x + 2}{x + 1} ).
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To express it as a single fraction with the denominator specified, multiply the numerator and denominator by ( x ), giving: [ \frac{3x^2 + 3x + 2}{x + 1} \times \frac{x}{x} = \frac{3x^3 + 3x^2 + 2x}{x(x + 1)} ]
Now we observe that: ( 3x^3 + 2x - 1 ) can be factored, leading to: [ \frac{3x^3 - 1}{x(x + 1)} ]
Therefore, from earlier expansions, this leads to a final simplification as ( \frac{3x - 1}{x} ). Thus: ( \frac{3x^3 - x - 2}{x^2 - 1} = \frac{3x - 1}{x(x + 1)} ).