The polynomial f(x) is defined by f(x) = 18x^3 + 3x^2 + 28x + 12. Use the Factor Theorem to show that (3x+2) is a factor of f(x).

The factor theorem states that if (x+a) is a factor of f(x), then f(-a)=0. This basically means that if (x+a) is a factor of f(x), then when (x+a)=0, f(x)=0, i.e. when x=-a, f(-a)=0. Our (x+a) in this case is (3x+2). If this is a factor, then when (3x+2)=0, f(x)=0. So when does (3x+2)=0? Well, this is a simple equation that we can rearrange. First, take 2 from both sides: 3x=-2. Now, divide by 3 on both sides: x=(-2/3). Now if x=(-2/3), then x=-a=-2/3, so a=2/3. Then f(-a) is f(-2/3), so if we replace each x with -2/3 in f(x), then this should equal zero if (3x+2) is a factor of f(x).Well, using our calculators we can quickly confirm that 18*(-2/3)^3 + 3*(-2/3)^2 + 28*(-2/3) + 12 = 0 as required. Hence (3x+2) is a factor of f(x) as required.