How do you find the product of (x+1)(x2+x+1)?

Question

traciaul6y · Accepted Answer

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

(x + 1) (x^{2} + x + 1)

becomes:

(x \times x^{2}) + (x \times x) + (x \times 1) + (1 \times x^{2}) + (1 \times x) + (1 \times 1)

x^{3} + x^{2} + x + x^{2} + x + 1

We can now group and combine like terms:

x^{3} + 1 x^{2} + 1 x + 1 x^{2} + 1 x + 1

x^{3} + 1 x^{2} + 1 x^{2} + 1 x + 1 x + 1

x^{3} + (1 + 1) x^{2} + (1 + 1) x + 1

x^{3} + 2 x^{2} + 2 x + 1

narodiloxe4 · Accepted Answer

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:

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=1
So we can start to write:

(x + 1) (x^{2} + x + 1) = x^{3}

x^{2}

: This can result from x*x or

18 x^{2}

, so the coefficient is:
1*1+1*1=2
So we can add

+ 2 x^{2}

to the result:

(x + 1) (x^{2} + x + 1) = x^{3} + 2 x^{2}

x^{1}

: This can result from x*1 or 1*x, so the coefficient is:
1*1+1*1=2
So we can add +2x to the result:

(x + 1) (x^{2} + x + 1) = x^{3} + 2 x^{2} + 2 x

...

x^{0}

: The constant term can only result from multiplying the constant term of the binomial by that of the trinomial, so:
1*1=1
So our final result is:

(x + 1) (x^{2} + x + 1) = x^{3} + 2 x^{2} + 2 x + 1

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

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

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