Question

Factor the polynomial completely given that k=4 is a zero. Also list the zeroes of P(x).

P(x) x^4-2x^3-7x^2-4x

Answer 1

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) $P(x)=x^4-2x^3-7x^2-4x=x(x^3-2x^2-7x-4)$

2) Consider the polynomial $x^3-2x^2-7x-4=x^3-4x^2+4x^2-2x^2-7x-4=x^2(x-4)+2x^2-7x-4=x^2(x-4)+2x^2-8x+8x-7x-4=x^2(x-4)+2x(x-4)+(x-4)=(x-4)(x^2+2x+1)$

3) The expression $x^2+2x+1$ is a perfect square:

$x^2+2x+1=(x+1)^2$

Therefore,

$P(x)=x(x-4)(x+1)^2.$

Answer: $P(x)=x(x-4)(x+1)^2.$ Zeroes: 0, 4, -1 (twice)