Let p(x) =30x^3 - 7x^2 -7x + 2. Prove that (2x+1) is a factor of p(x).

We use the factor theorem which states that (2x+1) is a factor of p(x) if and only if the root of (2x+1)=0 is also a root of p(x). The root of 2x+1=0 is x = -1/2 (-1 from both sides and then divide both sides by 2). Plugging this value into the polynomial we get the following30(-1/2)^3 - 7(-1/2)^2 - 7(-1/2) + 2.Resolving the powers we get30(-1/8) - 7(1/4) - 7(-1/2) + 2,which is equal to -30/8 - 7/4 + 7/2 + 2.Making all terms have a common denominator of 8 (4 would also work) we see that this is equivalent to-30/8 - 14/8 + 28/8 + 16/8This allows us to add the terms by adding terms in the numerator and see that (-1/2) is indeed a root of p(x)(-30-14+28+16)/8 = (-44+44)/8 = 0.We have shown p(-1/2)=0 which implies (2x+1) is a factor of p(x).