Multiply polynomial $ \, \color{blue}{ x-1 } \, $ by $ \, \color{orangered}{ x^3+x^2+x+1 }$.

$$ \left( \color{blue}{x-1} \right) \cdot \left( \color{orangered}{x^3+x^2+x+1} \right) = x^4-1 $$

We can multiply polynomials by using a GRID METHOD

Write one of the polynomials across the top and the other down the left side.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{1} \\ \hline \color{blue}{x} & & & & \\ \hline \color{blue}{-1} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{1} \\ \hline \color{blue}{x} & \color{orangered}{x^4} & \color{orangered}{x^3} & \color{orangered}{x^2} & \color{orangered}{x} \\ \hline \color{blue}{-1} & \color{orangered}{-x^3} & \color{orangered}{-x^2} & \color{orangered}{-x} & \color{orangered}{-1} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + x^3-x^3 + x^2-x^2 + x-x-1 = \\ x^4-1 $$

Polynomial Operations Calculator