Simplify,
\[ \frac{x^2 - 16}{x^2 - 3x - 4} \]
\[ (x + 3)(x - 4)(x + 5) \text{ is identical to } x^3 + ax^2 - 17x + b. \]
Find the value of a and the value of b.
Worked Solution & Example Answer:Simplify,
\[ \frac{x^2 - 16}{x^2 - 3x - 4} \]
\[ (x + 3)(x - 4)(x + 5) \text{ is identical to } x^3 + ax^2 - 17x + b - OCR - GCSE Maths - Question 17 - 2017 - Paper 1
Step 1
Simplify, \[ \frac{x^2 - 16}{x^2 - 3x - 4} \]
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Answer
To simplify the expression, we first factor the numerator and the denominator:
The numerator can be factored as:
x2−16=(x−4)(x+4)
The denominator can be factored as follows:
x2−3x−4=(x−4)(x+1)
Now, substituting the factored forms back in, we have:
(x−4)(x+1)(x−4)(x+4)
Canceling the common factor, we find:
x+1x+4
Thus, the simplified expression is x+1x+4.
Step 2
(x + 3)(x - 4)(x + 5) is identical to x^3 + ax^2 - 17x + b.
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Answer
To find the values of a and b:
First, expand the left side:
(x+3)(x−4)=x2−4x+3x−12=x2−x−12
Next, multiply by (x+5):
(x2−x−12)(x+5)=x3+5x2−x2−5x−12x−60
Simplifying this gives:
x3+4x2−17x−60
Comparing coefficients with x3+ax2−17x+b, we find:
![Simplify,--\[-\frac{x^2---16}{x^2---3x---4}-\]--\[-(x-+-3)(x---4)(x-+-5)-\text{-is-identical-to-}-x^3-+-ax^2---17x-+-b-OCR-GCSE Maths-Question 17-2017-Paper 1.png](https://cdn.simplestudy.io/assets/backend/uploads/question/subject_394_paper_3338_q17_67c001e8.jpg)
