Question

Using factor theorem, show that (x - 3) is a factor of x³ - 7x² + 15x - 9, Hence, factorise the given expression completely.

Answer 1

Let p(x) = x³ - 7x² + 15x - 9
For checking that (x - 3) is a factor of p(x), we find : p(3)
p(3) = (3)³ - 7(3)² + 15(3) - 9
= 27 - 63 + 45 - 9
= 72 - 72
= 0.
Hence, (x - 3) is a factor of p(x).
By division of p(x) by x - 3, we get the quotient
= x² - 4x + 3
∴ x³ - 7x² +15x - 9
= (x - 3) (x² - 4x + 3)
= (x - 3) (x - 3) (x - 1)
= (x - 3)² (x - 1).