How do you find the product of #(x + 1) (x^2 + x + 1)#?

2 Answers
Jun 13, 2017

Answer:

See a solution process below:

Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

#(color(red)(x) + color(red)(1))(color(blue)(x^2) + color(blue)(x) + color(blue)(1))# becomes:

#(color(red)(x) xx color(blue)(x^2)) + (color(red)(x) xx color(blue)(x)) + (color(red)(x) xx color(blue)(1)) + (color(red)(1) xx color(blue)(x^2)) + (color(red)(1) xx color(blue)(x)) + (color(red)(1) xx color(blue)(1))#

#x^3 + x^2 + x + x^2 + x + 1#

We can now group and combine like terms:

#x^3 + 1x^2 + 1x + 1x^2 + 1x + 1#

#x^3 + 1x^2 + 1x^2 + 1x + 1x + 1#

#x^3 + (1 + 1)x^2 + (1 + 1)x + 1#

#x^3 + 2x^2 + 2x + 1#

Jun 13, 2017

Answer:

#(x+1)(x^2+x+1) = x^3+2x^2+2x+1#

Explanation:

The way I like to do it is longer to explain than to do...

Look at each possible power of #x# in descending order and add up the different ways of getting it.

So in our example:

Given:

#(x+1)(x^2+x+1)#

we can tell that the highest possible power of #x# in the product is #3#, so work down from there:

#color(white)()#
#x^3#: This can only result from multiplying the #x# in the binomial by the #x^2# in the trinomial, so the coefficient is:

#1*1 = color(blue)(1)#

So we can start to write:

#(x+1)(x^2+x+1) = x^3...#

#color(white)()#
#x^2#: This can result from #x*x# or #1*x^2#, so the coefficient is:

#1*1+1*1 = color(blue)(2)#

So we can add #+2x^2# to the result:

#(x+1)(x^2+x+1) = x^3+2x^2...#

#color(white)()#
#x^1:# This can result from #x*1# or #1*x#, so the coefficient is:

#1*1+1*1 = color(blue)(2)#

So we can add #+2x# to the result:

#(x+1)(x^2+x+1) = x^3+2x^2+2x...#

#color(white)()#
#x^0:# The constant term can only result from multiplying the constant term of the binomial by that of the trinomial, so:

#1*1 = color(blue)(1)#

So our final result is:

#(x+1)(x^2+x+1) = x^3+2x^2+2x+1#

In practice (and with practice) the result line is all you need write: Adding up the coefficients can be done in your head.