What is the standard form of #y= (2x+3x^2)(x+3) -(x-2)^3#?

1 Answer
Dec 14, 2015

#y=2x^3 + 17x^2 - 6x + 8#


To answer this question, you will have to simplify the function. Begin by using the FOIL Method to multiply the first term:

#(2x+3x^2)(x+3) = 2x*x + 2x*3 + 3x^2*x +3x^2*3#

Simplifying this yields:


We now have the first term simplified. To simplify the second term, we can use the
Binomial Theorem, a useful tool when working with polynomials. One of the main points of the theorem is that the coefficients of an expanded binomial can be determined using a function called the choose function. The specifics of the choose function are more of a probability concept, so there's no need to go into it right now.

However, a simpler way to use the Binomial Theorem is
Pascal's Triangle. The numbers in Pascal's Triangle for a certain row number will correspond to the coefficients of the expanded binomial for that row number. In the case of cubing, the third row is #1,3,3,1#, so the expanded binomial would be:

#(a+b)^3 = 1a^3 + 3a^2b + 3ab^2 + 1b^3#

Notice how we decrease the power of #a# and increase the power of #b# as we move down the row. Evaluating this formula with the second term, #(x-2)^3#, yields:

#(x-2)^3 = x^3 + 3x^2(-2) + 3x(-2)^2+ (-2)^3#

Simplifying gives us:

#x^3 - 6x^2 +12x - 8#

To simplify, we can subtract the second term from the first:

#3x^3 + 11x^2 + 6x - (x^3 - 6x^2 +12x - 8) = 2x^3 + 17x^2 - 6x + 8#

Standard form means that the terms of the polynomial are ordered from highest degree to lowest. Because this has already been done, your final answer is:

#y = 2x^3 + 17x^2 - 6x + 8#