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If (x+1) is one factor of x^3 + 2x^2 + kx -6 find the other factors

Sagot :

Superkoopasirfidn
[tex]\text{Let } f(x) = x^3 + 2x^2 + kx - 6[/tex]

As (x + 1) is a factor, it follows that
[tex]f(-1) = 0[/tex]

[tex]\implies (-1)^3 + 2(-1)^2 + k(-1) - 6 = 0[/tex]

[tex]-1 + 2 -k - 6 = 0[/tex]

[tex]-k - 5 = 0[/tex]

[tex]k = -5[/tex]

[tex]\implies f(x) = x^3 + 2x^2 - 5x - 6[/tex]

Now we can use long division to find what is left of f(x) after it is divided by (x + 1). (Apologies, this is the best way I can represent long division on Brainly at this current time - I hope it's clear)

           x^2 +   x   -   6
x + 1 ( x^3 + 2x^2 - 5x - 6
           x^3 +   x^2
                       x^2 - 5x
                       x^2 + x
                               -6x - 6
                               -6x - 6
                                        0

So the remainder when f(x) is divided by (x + 1) is
[tex]x^2 + x - 6[/tex]

Factorising this we get
[tex](x + 3)(x - 2)[/tex]

So the three factors of f(x) are (x + 1), (x + 3) and (x - 2).
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