Find an expression for a cubic function f if f 4 160 and f 4 f 0 f 5 0 Step 1 A cubic function generally has the form f x ax bx2 cx d If we know that for some x value x p we have f p 0 then it must be true that x p is a factor of f x Since we are told that f 5 0 we know that x 5 is a factor Step 2 Similarly since f 4 0 then f x has the factor x 4 I to obtain a Step 3 Since f x has the factors x x 5 and x 4 then we must have f x a x x 5 x 4 for some as yet unknown multiplier a However since it was given that f 4 160 we substitute 4 for x and 160 for f x and solve f 4 160 a 4 4 5 4 4 f x 5 Step 4 Using all the information we found we can now write the full expression x 4 and since f 0 0 then f x has the factor