How do I factor 18x^3-15x^2-4x+4?

Question

This is for a project my teacher told me to factor it by using last over first, but I don't know how to do that.

Answer 1

I think that by "last over first", your teacher means to use the Rational Zero Theorem, which says that if p/q is a rational zero of a polynomial. then p is a divisor of the constant term (last)and q is a divisor of the leading coefficient (first). 

 

So, in this problem, p is a divisor of 4 and q is a divisor of 18

 

p = 1, -1, 2, -2, 4, -4

q = 1, -1, 2, -2, 3, -3, 6, -6, 9, -9, 18, -18

 

Therefore, p/q = 1, -1, 2, -2, 4, -4, 1/3, -1/3, 2/3, -2/3, 4/3, -4/3, 2/9, -2/9, 4/9, -4/9, 1/9, -1/9, 1/18, -1/18 

 

By testing the various possibilities, we see that -1/2 is a zero.  

 

Divide synthetically by x - (-1/2):

 

-1/2⌋     18       -15      -4       4

         ________      -9____12____-4

            18       -24       8       0

 

18x³ -15x² - 4x + 4 = (x+½)(18x²-24x+8)

 

                            = (2x+1)(9x²-12x+4)

 

                            = (2x+1)(3x-2)(3x-2)

 

Setting each factor to 0 we get x = -1/2 and 2/3.