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Algebra
Factorisation Factorize the following:
1. x^3 – 4 x^2 + 5 x – 2 Solution: 1 – 4 + 5 – 2 = 0 ( x 1) is a factor 1 1 4 5 2 0 1 3 2 1 3 2 0 x 2 - 3 x + 2 = ( x – 1) ( x – 2) The factors are ( x – 1) ( x – 2) ( x – 3) = ( x – 1) 2 ( x – 2) Factorize the following: 2. x **3
Solution: 1 + 9 + 23 + 15 0 ( x 1) is not a factor
1 + 23 = 9 + 15 = 24 ( x 1) is a factor 1 1 9 23 15 0 1 8 15 1 8 15 0 x^2 + 8 x + 15 = ( x + 3) ( x + 5)
The factors are ( x + 1) ( x + 3) ( x + 5)
3. x^3 2 x^2 5 x + 6 Solution: 1 2 5 + 6 = 0 ( x 1) is a factor 1 5 2 + 6 ( x + 1) is not a factor 1 1 2 5 6 0 1 1 6 1 1 6 0 x 2 x 6 = ( x 3) ( x + 2)
The factors are ( x 1) ( x 3) ( x + 2)
Factorize the following:
4. 2 x^4 + 7 x^3 + x^2 – 7 x 3 Solution: 2 + 7 + 1 7 3 = 0 ( x 1) is a factor 2 + 1 3 = 7 7 = 0 ( x + 1) is also a factor 1 2 7 1 7 3 0 2 9 10 3 1 2 9 10 3 0 0 2 7 3 2 7 3 0 2 x 2
- 7 x + 3 = 2 x 2
- 6 x + x + 3 = 2 x ( x + 3) + 1 ( x + 3) = ( x + 3) (2 x + 1)
The factors are ( x 1) ( x + 1) ( x + 3) ( 2 x + 1)
7. m 3 - 2 m 2 - 4 m + 8 Solution:
1 2 4 + 8 0 ( m 1) is not a factor
1 4 2 + 8 ( m + 1) is not a factor
In that case try m = 2 2 1 2 4 8 0 2 0 8 1 0 4 0 m^2 4 = ( m + 2) ( m 2)
The factors are ( m 2)^2 ( m + 2)
Factorize the following:
8. a^3 – 5 a^2 – 2 a + 24 Solution: 1 5 2 + 24 0 ( a 1) is not a factor 1 2 5 + 24 1 ( a + 1) is also not a factor Try a = 2 2 1 5 2 24 0 2 6 16 1 3 8 8 ( a 2) is not a factor Try a = 2 2 1 5 2 24 0 2 14 24 1 7 12 0 a^2 – 7 a + 12 = ( a 3) ( a 4) The factors are ( a + 2) ( a 3) ( a 4)
9. x 4 5 x + 4 Solution: x 4
- 5 x + 0 x + 4 1 + 0 – 5 + 0 + 4 = 0 ( x 1) is a factor 1 5 + 4 = 0 ( x + 1) is also a factor 1 1 0 5 0 4 0 1 1 4 4 1 1 1 4 4 0 0 1 0 4 1 0 4 0 x 2 4 = ( x 2) ( x + 2) The factors are ( x 2) ( x + 2) ( x 1) ( x + 1)
