How do you list all possible roots and find all factors and zeroes of $x^{3} + 4 x^{2} + 5 x + 2$ $x^3+4x^2+5x+2$ ?

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Answer 1 · 2016-09-01T05:36:45

$p (x) = x^{3} + 4 x^{2} + 5 x + 2 = {(x + 1)}^{2} (x + 2)$ $p(x)=x^3+4x^2+5x+2=(x+1)^2(x+2)$ .

The roots of $p(x)=0, &, the zeroes of p(x) are, - 1, - 1, - 2$ $"p(x)=0, &, the zeroes of p(x) are, "-1,-1,-2$ .

Name the given poly. $p (x) = x^{3} + 4 x^{2} + 5 x + 2$ $p(x)=x^3+4x^2+5x+2$ .

We notice that,

$The sum of the co-effs. of odd-powered terms = 1 + 5 = 6, &,$ $"The sum of the co-effs. of odd-powered terms"=1+5=6, &,$

$The sum of the co-effs. of even-powered terms = 4 + 2 = 6 .$ $"The sum of the co-effs. of even-powered terms"=4+2=6.$

This means that $x + 1$ $x+1$ is a factor of $p (x)$ $p(x)$ .

$Now p(x) = x^{3} + 4 x^{2} + 5 x + 2$ $"Now p(x)"=x^3+4x^2+5x+2$ ,

$=ul(x^3+x^2)+ul(3x^2+3x)+ul(2x+2)$ ,

$=x^2(x+1)+3x(x+1)+2(x+1)$ ,

$=(x+1)(x^2+3x+2)$ ,

$=(x+1){ul(x^2+2x)+ul(x+1)}$ ,

$=(x+1){x(x+1)+1(x+1)}$ ,

$=(x+1){(x+1)(x+2)}$ ,

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

Clearly, the roots of $"p(x)=0, &, the zeroes of p(x) are, "-1,-1,-2$ .

How do you list all possible roots and find all factors and zeroes of x^3+4x^2+5x+2x3+4x2+5x+2x^3+4x^2+5x+2?