So we have to write the following polynomial expression as a product of two factors:
[tex]P(x)=2x^3+5x^2+5x+6[/tex]
In order to do this we should find one of its roots first i.e. a x value that makes P(x)=0. If we use r to label this root we can write P like:
[tex]P(x)=(x-r)\cdot(ax^2+bx+c)[/tex]
Where a, b and c are numbers that we can find using Ruffini's rule. So first of all let's find a root. We can use the rational root theorem. It states that if P(x) has rational roots then they are given by the quotient between a factor of the constant term (i.e. the number not multplied by powers of x) and a factor of the leading coefficient (i.e. the number multiplying the biggest power of x). In this case we have to look for the factors of 6 and 2 respectively. Their factors are:
[tex]\begin{gathered} 6\colon6,-6,3,-3,2,-2,1,-1 \\ 2\colon2,-2,1,-1 \end{gathered}[/tex]
And the quotients and possible values for r are:
[tex]6,-6,3,-3,2,-2,\frac{3}{2},-\frac{3}{2},1,-1,\frac{1}{2},-\frac{1}{2}[/tex]
So one of these numbers make P(x) equal to zero. For example if we take x=-2 we get:
[tex]\begin{gathered} P(-2)=2\cdot(-2)^3+5\cdot(-2)^2+5\cdot(-2)+6 \\ P(-2)=-16+20-10+6=0 \end{gathered}[/tex]
So -2 is a root of P(x) which means that we can take r=-2.
Now we can use Ruffini's law. On the first row we write the coefficients of P(x). Then the first one is repeated in the third row:
Now we multiply 2 by -2 and we write the result under the second coefficient. Then we add them:
Now we do the same with the 1:
And then we multiply 3 and -2 and add the result ot the last coefficient:
The numbers 2, 1 and 3 are the values of a,b and c respectively. Then we can write P(x) as a product of two factors and the answer is:
[tex]P(x)=(x+2)(2x^2+x+3)[/tex]
