1)Given xReturn to the tutorial^{3}- 2x^{2}- 5x + 6 with one factor being (x + 2), find the remaining factors. SOLUTION: If (x + 2) is a factor, then (x + 2) must divide x^{3}- 2x^{2}- 5x + 6 exactly (i.e. there are no remainders!!) Set up the problem using long division: ----- ------------------ (note that this | sign is the (x + 2)| x^{3}- 2x^{2}- 5x + 6 one often used for long division) STEP 1: Look at only the first term, x, in the given factor (x + 2) and see what value, when multiplied by this term will give you the first term in the expression being divided. In this case, the value would be x^{2}because x^{2}(x) = x^{3}. Put this value in top of the line just like in regular division. It now looks like this: x^{2}------------------- (x + 2)| x^{3}- 2x^{2}- 5x + 6 STEP 2: Multiply this value that is now on top by the whole factor you are dividing by and write it below the expression being divided (i.e. x^{2}(x + 2) = x^{3}+ 2x^{2}) Thus: x^{2}------------------- (x + 2)| x^{3}- 2x^{2}- 5x + 6 x^{3}+ 2x^{2}STEP 3: Subtract this new expression from hte original expression and write the answer on the next line. Bring down the next term from the original expression and write it next to your answer. It should look like: x^{2}------------------- (x + 2)| x^{3}- 2x^{2}- 5x + 6 x^{3}+ 2x^{2}--------- -4x^{2}- 5x STEP 4: Repeat Steps 1-3 again with the expression of interest now being (-4x^{2}- 5x). It should end up looking like this: x^{2}- 4x + 3 ------------------- (x + 2)| x^{3}- 2x^{2}- 5x + 6 x^{3}+ 2x^{2}--------- -4x^{2}- 5x -4x^{2}- 8x --------- 3x + 6 3x + 6 ------ 0 So the other factor of x^{3}- 2x^{2}- 5x + 6 besides (x + 2) is (x^{2}- 4x + 3). But this factor can be factored further into (x - 1) and (x - 3) (by decomposition). Thus, x^{3}- 2x^{2}- 5x + 6 = (x + 2)(x - 1)(x - 3) 2) Given ( x - 2) as a factor, find the other factor of x^{3}- 2x^{2}+ 3x - 6. SOLUTION: x^{2}+ 0x + 3 ------------------ (x - 2)| x^{3}- 2x^{2}+ 3x - 6 x^{3}- 2x^{2}-------- 0x^{2}+ 3x 0x^{2}- 0x --------- 3x - 6 3x - 6 ------ 0 The other factor is then (x^{2}+ 3) [omit the 0x term].