To multiply: the given binomial using FOIL method. \left(\frac{

varaderiyw 2021-11-23 Answered
To multiply: the given binomial using FOIL method.
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}\)

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Expert Answer

Sevensis1977
Answered 2021-11-24 Author has 142 answers
Step 1
FOIL Method: it is a mnemonic for the standard method of multiplying of two binomials.
F stands for the product of first terms.
O stands for the product of outer terms.
I stand for the product of inner terms.
L stands for the product of last terms.
The given equation is
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}\)
Now, multiply the given binomial usinf FOIL method,
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}+{b}^{{{2}}}\right)}\)
\(\displaystyle={\left({\frac{{{2}}}{{{3}}}}{a}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}\right)}+{\left({\frac{{{2}}}{{{3}}}}{a}\right)}{\left({b}^{{{2}}}\right)}+{\left(-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}\right)}+{\left({b}^{{{2}}}\right)}{\left(-{b}^{{{2}}}\right)}\)
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}\)
\(\displaystyle={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}+{\frac{{{2}}}{{{3}}}}{a}{b}^{{{2}}}-{\frac{{{2}}}{{{3}}}}{a}{b}^{{{2}}}-{b}^{{{4}}}\)
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}-{b}^{{{4}}}\)
Hence, the multiplication of the given binomial is
\(\displaystyle{\left({\frac{{{2}}}{{{3}}}}{a}-{b}^{{{2}}}\right)}{\left({\frac{{{2}}}{{{3}}}}{a}^{{{2}}}+{b}^{{{2}}}\right)}={\frac{{{4}}}{{{9}}}}{a}^{{{2}}}-{b}^{{{4}}}\)
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