Question

Explain how to use the distributive property to find the product. $$ (x+1)\left(x^{2}-x+1\right) $$

Short answer

The product of the binomial \(x+1\) and the trinomial \(x^{2}-x+1\) is \(x^{3} + 1\).

Step-by-step solution

Distribute the First Term of the Binomial

First, distribute the 'x' across each term in the trinomial. This gives: \(x(x^{2}) - x(x) + x(1) = x^{3} - x^{2} + x\)

Distribute the Second Term of the Binomial

Next, distribute the '1' across each term in the trinomial. This results in: \(1(x^{2}) - 1(x) + 1(1) = x^{2} - x + 1\).

Combine Like Terms

Finally, we must combine the computed terms. When combined, we get: \(x^{3} - x^{2} + x + x^{2} - x + 1 = x^{3} +1\).