Answer:
P(x)=(x-2)(x-4)(x+3)(x+6)
Step-by-step explanation:
Given: P(x)=x⁴+3x³-28x²-36x+144
It is a polynomial with degree 4.
It should maximum four factor.
Hit and trial error method.
Put x = 2 into P(x)
P(2)=2⁴+3×2³-28×2²-36×2+144
P(2) = 0
So, x-2 would be factor of P(x)
Now divide x⁴+3x³-28x²-36x+144 by x-2 to get another factors
[tex](x^4+3x^3-28x^2-36x+144)\div (x-2) = x^3+5x^2-18x-72[/tex]
[tex]P(x)=(x-2)(x^3+5x^2-18x-72)[/tex]
Put x = 4
[tex]P(4) = 0 [/tex]
now divide [tex]x^3+5x^2-18x-72[/tex] by x-4
[tex](x^3+5x^2-18x-72)\div (x-4) = x^2+9x+18[/tex]
[tex]P(x)=(x-2)(x-4)(x^2+9x+18)[/tex]
Now factor [tex]x^2+9x+18[/tex]
[tex]\Rightarrow x^2+9x+18[/tex]
[tex]\Rightarrow (x+6)(x+3)[/tex]
Complete factor of P(x)
P(x)=(x-2)(x-4)(x+3)(x+6)
