Respuesta :
Answer:
The factors are x-2,[tex]x-\frac{2}{3}[/tex] and x+4
Therefore the expression can be written as [tex]3x^3+7x^2-18x+8=(x-2)(x-\frac{2}{3})(x+4)[/tex]
Step-by-step explanation:
Given expression is [tex]3x^3+7x^2-18x+8[/tex]
And also given that x-1 is one of the factors
i.e.,x-1=0
x=1
To find the factors equate the given expression to zero
[tex]3x^3+7x^2-18x+8=0[/tex]
Using synthetic division to find the factors
1_| 3 7 -18 8
0 3 10 -8
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3 10 -8 0
Therefore the quadratic equation is [tex]3x^2+10x-8=0[/tex]
To find the factors of the above equation:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Where a and b are coefficients of [tex]x^2[/tex] and x respectively
[tex]x=\frac{-10\pm\sqrt{10^2-4(3)(-8)}}{2(3)}[/tex] Where a=3 , b=10 and c=-8
[tex]=\frac{-10\pm\sqrt{100+96}}{6}[/tex]
[tex]=\frac{-10\pm\sqrt{196}}{6}[/tex]
[tex]=\frac{-10\pm14}{6}[/tex]
[tex]x=\frac{-10\pm14}{6}[/tex]
Therefore [tex]x=\frac{-10+14}{6}[/tex] and [tex]x=\frac{-10-14}{6}[/tex]
[tex]x=\frac{4}{6}[/tex] and [tex]x=\frac{-24}{6}[/tex]
Therefore [tex]x=\frac{2}{3}[/tex] and [tex]x=-4[/tex]
Therefore the factors are x-2,[tex]x-\frac{2}{3}[/tex] and x+4
[tex]3x^3+7x^2-18x+8=(x-2)(x-\frac{2}{3})(x+4)[/tex]
