If it is known that k=4 is polynomial's zero, then the polynomial should have x-4 as its factor.
Factor the polynomial:
1) [tex]P(x)=x^4-2x^3-7x^2-4x=x(x^3-2x^2-7x-4)[/tex]
2) Consider the polynomial [tex]x^3-2x^2-7x-4=x^3-4x^2+4x^2-2x^2-7x-4=x^2(x-4)+2x^2-7x-4=x^2(x-4)+2x^2-8x+8x-7x-4=x^2(x-4)+2x(x-4)+(x-4)=(x-4)(x^2+2x+1)[/tex]
3) The expression [tex]x^2+2x+1[/tex] is a perfect square:
[tex]x^2+2x+1=(x+1)^2[/tex]
Therefore,
[tex]P(x)=x(x-4)(x+1)^2.[/tex]
Answer: [tex]P(x)=x(x-4)(x+1)^2.[/tex] Zeroes: 0, 4, -1 (twice)
