Question

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

Answer 1

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`

Answer 2

`(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.