Question 1123899
if f(-1) = 0, then one of the factors has to be x+1.


that's because, if x + 1 = 0, then solving for x will get x = -1.


your equation is f(x) = 6x^3 + 13x^2 + 2x -5.


f(-1) is solved by replacing x with -1 and evaluating the expression.


f(-1) becomes 6 * (-1)^3 + 13 * (-1)^2 + 2 * (-1) - 5.


the result is -6 + 13 - 2 - 5 = 0.


therefore, you need to divide 6x^3 + 13x^2 + 2x - 5 by (x+1).


alternatively, you can use synthetic division with -1 operating on the coefficients of 6x^3 + 13x^2 + 2x - 5.


either way, you'll get 6x^3 + 13x^2 + 2x - 5 divided by (x + 1) = 6x^2 + 7x - 5.


you can factor 6x^2 + 7x - 5 and get the factors of (2x - 1) * (3x + 5)


therefore, the factors of 6x^3 + 13x^2 + 2x - 5 are (x + 1) * (2x - 1) * (3x + 5).


it can be seen graphically that these are equivalent expressions because both equations will draw the same line, as shown below.


<img src = "http://theo.x10hosting.com/2018/092202.jpg" alt="$$$" >


my calculations in the long division and the synthetic division are shown below.


the top section calculations are long division.
the bottom section calculations are synthetic division.


<img src = "http://theo.x10hosting.com/2018/092203.jpg" alt="$$$" >


if you don't know how to do long division or synthetic division or how to factor quadratic equations, here's some references that should help you.


<a href = "https://www.purplemath.com/modules/polydiv2.htm" target = "_blank">https://www.purplemath.com/modules/polydiv2.htm</a>


<a href = "https://www.purplemath.com/modules/synthdiv.htm" target = "_blank">https://www.purplemath.com/modules/synthdiv.htm</a>


<a href = "https://www.purplemath.com/modules/factquad.htm" target = "_blank">https://www.purplemath.com/modules/factquad.htm</a>


when factoring quadratics, if all else fails, use the quadratic formula.


that formula is:


<pre>


             x =    -b plus or minus sqrt(b^2 - 4ac)
                  -------------------------------------
                                 2a

</pre>


for example, if it tells you that x = -1, then you want that equation to be equal to 0.
you would add 1 to both sides of that equation to get x + 1 = 0.
that means (x + 1) is one of your factors.