Answer :
Answer:
The factors are (x+6),[tex]x-2i\sqrt{2}[/tex] and [tex]x+2i\sqrt{2}[/tex]
Therefore the given polynomial
[tex]x^3+6x^2+8x+48=(x+6)(x-2i\sqrt{2})(x+2i\sqrt{2})[/tex]
Step-by-step explanation:
Given polynomial is [tex]x^3+6x^2+8x+48[/tex]
To factorise the polynomial equate the given polynomial to zero
[tex]x^3+6x^2+8x+48=0[/tex]
By using synthetic division we can find the factors :
[tex]x^3+6x^2+8x+48=0[/tex]
_-6| 1 6 8 48
0 -6 0 -48
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1 0 8 0
Therefore (x+6) is a factor
The quadratic equation is [tex]x^2+8=0[/tex]
To solve it we can use [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Here a=1 b=0 and c=8
[tex]x=\frac{-(0)\pm\sqrt{0^2-4(1)(8)}}{2(1)}[/tex]
[tex]=\frac{\pm\sqrt{-32}}{2}[/tex]
[tex]=\frac{\pm\sqrt{32i^2}}{2}[/tex] where [tex]i^2=-1[/tex]
[tex]=\frac{\pm4i\sqrt{2}}{2}[/tex]
[tex]=\pm2i\sqrt{2}[/tex]
[tex]x=\pm2i\sqrt{2}[/tex]
Therefore [tex]x=2i\sqrt{2}[/tex] and [tex]x=-2i\sqrt{2}[/tex]
Therefore the factors are (x+6),[tex]x-2i\sqrt{2}[/tex] and [tex]x+2i\sqrt{2}[/tex]
Therefore the given polynomial
[tex]x^3+6x^2+8x+48=(x+6)(x-2i\sqrt{2})(x+2i\sqrt{2})[/tex]
