Answer:
The factor form of given expression is (x-4)(x-2i)(x+2i).
Step-by-step explanation:
The given expression is
[tex]x^3-4x^2+4x-16[/tex]
It can be written as
[tex]f(x)=x^3-4x^2+4x-16[/tex]
According to the rational root theorem, all possible rational roots are in the form of
[tex]\frac{a_0}{a_n}[/tex]
Where, a₀ is constant term and [tex]a_n[/tex] is leading coefficient.
[tex]f(4)=4^3-4(4)^2+4(4)-16=0[/tex]
Since the value of f(x) is 0 at x=4, therefore 4 is a root of the function and (x-4) is a factor of given expression.
Use synthetic method to find the remaining factors.
[tex](x^3-4x^2+4x-16)=(x-4)(x^2+4)[/tex]
[tex](x^3-4x^2+4x-16)=(x-4)(x^2-(2i)^2)[/tex]
[tex](x^3-4x^2+4x-16)=(x-4)(x-2i)(x+2i)[/tex]
Therefore the factor form of given expression is (x-4)(x-2i)(x+2i).
Answer:
(x-4) (x-2i) (x+2i)
Step-by-step explanation:
x^3−4x^2+4x−16
I will factor by grouping
x^3−4x^2+ 4x−16
The first group is the first 2 terms. I can factor out an x^2
x^2 (x-4) + 4x-16
The second group I can factor out a 4
x^2 (x-4) + 4(x-4)
Now I can factor out (x-4)
(x-4) (x^2 +4)
We know that x^2 + 4 factors into +2i and -2i because
(a^2 -b^2) = (a-b) (a+b) but we have (a^2 + b^2) = (a-bi) (a+bi)
(x-4) (x-2i) (x+2i)
