Question

# To find: the square of a polynomial (4a-3b)^{2} using indicated operation.

Polynomial arithmetic
To find: The square of a polynomial $$\displaystyle{\left({4}{a}-{3}{b}\right)}^{{{2}}}$$ using indicated operation.

2021-07-31
Concept:
Multiplication of polynomials:
1) To find the product of two polynomials, distribute each term of one polynomial, multiplying by each term of other polynomial.
2) Another method is to write such a product vertically, similar to the method used in arithmetic for multiplying whole numbers.
3) The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outer, Inner, Last) gives the pairs of terms to be multiplied when distributing each term of the first binomial, multiplying by each term of the second binomial.
4) Using the Special Products like
Square of a Binomial: $$\displaystyle{\left({a}+{b}\right)}^{{{2}}}={a}^{{{2}}}+{2}{a}{b}+{b}^{{{2}}}$$
and $$\displaystyle{\left({a}-{b}\right)}^{{{2}}}={a}^{{{2}}}-{2}{a}{b}+{b}^{{{2}}}$$
Product of the Sum and Difference of Two Terms: $$\displaystyle{a}^{{{2}}}-{b}^{{{2}}}={\left({a}+{b}\right)}{\left({a}-{b}\right)}$$
Calculation:
Given that $$\displaystyle{\left({4}{a}-{3}{b}\right)}^{{{2}}}$$
To find the product of two, apply Square of a Binomial $$\displaystyle{\left({a}-{b}\right)}^{{{2}}}={a}^{{{2}}}-{2}{a}{b}+{b}^{{{2}}}$$
$$\displaystyle={\left({4}{a}\right)}^{{{2}}}-{2}{\left({4}{a}\right)}{\left({3}{b}\right)}+{\left({3}{b}\right)}^{{{2}}}$$
$$\displaystyle={16}{a}^{{{2}}}-{24}{a}{b}+{9}{b}^{{{2}}}$$ [Expand exponentials by power rule 2 and multiply]
Final statement:
Therefore, $$\displaystyle{\left({4}{a}-{3}{b}\right)}^{{{2}}}={16}{a}^{{{2}}}-{24}{a}{b}+{9}{b}^{{{2}}}$$