Show that (x + 1)(x + 3)(x + 5) can be written in the form ax^3 + bx^2 + c^x + d where a, b, c and d are positive integers.
The way to go about a question like this is to break it down into steps; rather than trying to multiply all the brackets at once expand two first then multiply by the third. -----------------------------------------------------------------------E.g. Q. Show (x+1)(x+3)(x+5) = ax3+bx2+cx+d-----------------------------------------------------------------------Step 1. Expand two brackets, in this case I will use (x+3) and (x+5) as this will leave (x+1) to multiply the resultant quadratic expression by-For the expansion you can solve it on inspection or another method is the FOIL, First - Outside - Inside - Last, method; which will be very useful for the second expansion. (x+3)(x+5) --- First - xx = x2 ----- Outside - x(+)5 =(+) 5x ----- Inside - x*(+)3 = (+)3x ----- Last - (+)3*(+)5 = (+)15 (Note the signs in brackets, I would recommend putting them in as you can easily get caught out. If seen in your working that you were using the right sign and then accidently switched you're more likely to get some marks for working out rather than none for just a wrong answer)Now add together to get (x2+8x+15) which you can now expand in step 2 -----------------------------------------------------------------------Step 2. Expand your quadratic expression from Step 1 with the final bracket(x2+8x+15)(x+1). Using the FOIL method (this time with two insides) again you get x2 * x = x3 ----- x2 * (+)1 =(+) x2 ----- (+)8x*(+)x= (+)8x2 ----- (+)15 * (+)x=(+)15x ----- (+)15*(+)1=(+)15When all like terms are combined and simplified you get the expression (x3+9x2+15x+15)Which is in the form you are asked to find, where a= 1, b=9, c=15 and d=15 (you could write this on the side to make it more explicit)With more practice and experience expanding brackets you will be able to take out the individual FOIL steps and the signage however since you can write these steps as you do the calculations I would recommend keeping them so you or an examiner can work through it if something doesn't make sense
