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17 (a) Simplify: \( \frac{x^2 - 16}{x^2 - 3x - 4} \) (b) \( (x + 3)(x - 4)(x + 5) \) is identical to \( x^3 + ax^2 - 17x + b \) - OCR - GCSE Maths - Question 17 - 2017 - Paper 1
Question 17
17 (a) Simplify: \( \frac{x^2 - 16}{x^2 - 3x - 4} \) (b) \( (x + 3)(x - 4)(x + 5) \) is identical to \( x^3 + ax^2 - 17x + b \). Find the value of a and the value... show full transcript
Worked Solution & Example Answer:17 (a) Simplify: \( \frac{x^2 - 16}{x^2 - 3x - 4} \) (b) \( (x + 3)(x - 4)(x + 5) \) is identical to \( x^3 + ax^2 - 17x + b \) - OCR - GCSE Maths - Question 17 - 2017 - Paper 1
Step 1
Simplify: \( \frac{x^2 - 16}{x^2 - 3x - 4} \)
Answer
To simplify the expression, we first factor both the numerator and the denominator.
The numerator (x^2 - 16) is a difference of squares:
Next, we factor the denominator (x^2 - 3x - 4):
To factor this, we look for two numbers that multiply to (-4) and add to (-3) which are (-4) and (1):
Now, we can rewrite the expression as:
Now, we can cancel the common factor (x - 4) (as long as (x eq 4)):
Thus, the simplified form is:
Step 2
Find the value of a and the value of b.
Answer
To find the values of (a) and (b), we first expand the expression ((x + 3)(x - 4)(x + 5)):
- Expand ((x + 3)(x - 4) ): [(x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12]
- Now, multiply this result by ((x + 5)): [(x^2 - x - 12)(x + 5) = x^3 + 5x^2 - x^2 - 5x - 12x - 60] Combine like terms: [= x^3 + (5 - 1)x^2 + (-5 - 12)x - 60 = x^3 + 4x^2 - 17x - 60]
From the expression (x^3 + ax^2 - 17x + b), we can compare coefficients:
- The coefficient of (x^2) gives us (a = 4).
- The constant term gives us (b = -60).
Thus, the values are: [ a = 4 ] [ b = -60 ]