Question

The factrisation of x³ - 3x² - 9x - 5 is _____
Options:-

(x + 1)² (x - 5)

(x - 1)² (x - 5)

(x² - 1) (x - 5)

(x² - 1) (x + 5)

Answer 1

Step-by-step explanation:

We have,,

$\sf{x^{3}-3x^{2}-9x-5}$

First check whether x+1 is a factor of the given polynomial,

we have, $\sf{p(x)=x^{3}-3x^{2}-9x-5}$

$\implies{\sf{p(-1)=(-1)^{3}-3(-1)^{2}-9(-1)-5}}$

$\implies{\sf{p(-1)=-1-3+9-5}}$

$\implies{\sf{p(-1)=0}}$

Since result is zero therefore x+1 is a factor of the given polynomial,

Now,,

$\sf{x^{3}-3x^{2}-9x-5}$

$\sf{=x^{3}+x^{2}-4x^{2}-4x-5x-5}$

$\sf{=x^{2}(x+1)-4x(x+1)-5(x+1)}$

$\sf{=(x+1)[x^{2}-4x-5]}$

$\sf{=(x+1)[x^{2}-5x+x-5]}$

$\sf{=(x+1)[x(x-5)+1(x-5)]}$

$\sf{=(x+1)[(x+1)(x-5)]}$

$\sf{=(x+1)(x+1)(x-5)}$

$\sf{=(x+1)^{2}(x-5)}$

Therefore, option (i) is correct...✓✓✓

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