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f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 3
Question 5
f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x). (b) Factorise f(x) completely.
Worked Solution & Example Answer:f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 3
Step 1
Use the factor theorem to show that (x + 2) is a factor of f(x).
Answer
To use the factor theorem, we substitute the value that makes the factor zero. For the factor (x + 2), the zero is x = -2.
Calculating f(-2):
Calculating each term:
- Constant term:
Putting it all together:
Calculating further:
Since f(-2) = 0, by the factor theorem, (x + 2) is indeed a factor of f(x).
Step 2
Factorise f(x) completely.
Answer
Starting from the polynomial:
Since we know that (x + 2) is a factor, we can factor f(x) using synthetic division or polynomial long division.
Using synthetic division:
-2 | 2 -7 -10 24
| -4 22 -16 |
|---|
2 -11 12 0
The result is:
Now we need to factor the quadratic part, 2x² - 11x + 12. The factors of this can be found by looking for two numbers that multiply to (2*12) = 24 and add to -11. These numbers are -3 and -8. Rewrite the quadratic:
Factor by grouping:
Thus, we have:
This is the complete factorization of f(x).