Question

Using factor theorem, factorise the following:

\[x^{3} + 7x^{2} + 7x - 15\]

Answer 1

Let $f(x) = x^3 + 7x^2 + 7x - 15$. 

By the factor theorem, $(x - a)$ is a factor of $f(x)$ if $f(a) = 0$. 

Since $f(1) = 1 + 7 + 7 - 15 = 0$, $(x - 1)$ is a factor. 

Dividing $$f(x)$$ by $$(x - 1)$$ gives $$x^2 + 8x + 15$$. 

Hence, $$x^3 + 7x^2 + 7x - 15$$

$$= (x - 1)(x^2 + 8x + 15)$$

$$= (x - 1)(x + 3)(x + 5)$$