\(\displaystyle{\left({x}-{1}\right)}^{{2}}={x}^{{2}}-{2}{x}+{1}\)

Explanation:

FOIL is a mnemonic to help remember which combinations of terms need to be combined to evaluate the product of two binomials.

In our example, \(\displaystyle{\left({x}-{1}\right)}^{{2}}={\left({x}-{1}\right)}{\left({x}-{1}\right)}\) can be evaluated as follows:

\(\displaystyle{\left({x}-{1}\right)}{\left({x}-{1}\right)}=\overbrace{{{x}\cdot{x}}}^{{\text{First}}}+\overbrace{{{x}\cdot{\left(-{1}\right)}}}^{{\text{Outside}}}+\overbrace{{{\left(-{1}\right)}\cdot{x}}}^{{\text{Inside}}}+\overbrace{{{\left(-{1}\right)}\cdot{\left(-{1}\right)}}}^{{\text{Last}}}\)

\(\displaystyle={x}^{{2}}-{x}-{x}+{1}\)

\(\displaystyle={x}^{{2}}-{2}{x}+{1}\)

Explanation:

FOIL is a mnemonic to help remember which combinations of terms need to be combined to evaluate the product of two binomials.

In our example, \(\displaystyle{\left({x}-{1}\right)}^{{2}}={\left({x}-{1}\right)}{\left({x}-{1}\right)}\) can be evaluated as follows:

\(\displaystyle{\left({x}-{1}\right)}{\left({x}-{1}\right)}=\overbrace{{{x}\cdot{x}}}^{{\text{First}}}+\overbrace{{{x}\cdot{\left(-{1}\right)}}}^{{\text{Outside}}}+\overbrace{{{\left(-{1}\right)}\cdot{x}}}^{{\text{Inside}}}+\overbrace{{{\left(-{1}\right)}\cdot{\left(-{1}\right)}}}^{{\text{Last}}}\)

\(\displaystyle={x}^{{2}}-{x}-{x}+{1}\)

\(\displaystyle={x}^{{2}}-{2}{x}+{1}\)