Special Rules of Multiplication

92. Special Rules of Multiplication.

Some results of multiplication are of so great utility in shortening algebraic work that they should be carefully noticed and remembered. The following are important:

93. Square of the Sum of Two Numbers.

$$\begin{gathered} (a+b{)}^2\&=(a+b)(a+b) \\ \&=a(a+b)+b(a+b) \\ \&=a^2+ab+ab+b^2 \\ \&=a^2+2ab+b^2. \end{gathered}$$

Since $a$ and $b$ stand for any two numbers, we have

The square of the sum of two numbers is the sum of their squares plus twice their product.

94. Square of the Difference of Two Numbers.

$$\begin{gathered} (a-b{)}^2\&=(a-b)(a-b) \\ \&=a(a-b)-b(a-b) \\ \&=a^2-ab-ab+b^2 \\ \&=a^2-2ab+b^2. \end{gathered}$$

Hence we have

The square of the difference of two numbers is the sum of their squares minus twice their product.

95. Product of the Sum and Difference of Two Numbers.

$$\begin{gathered} (a+b)(a-b)\&=a(a-b)+b(a-b) \\ \&=a^2-ab+ab-b^2 \\ \&=a^2-b^2. \end{gathered}$$

Hence, we have

The product of the sum and difference of two numbers is the difference of their squares.

If we put $2x$ for $a$, and $3$ for $b$, we have

Exercise 26.

Write by inspection the value of:

1. $(m+n{)}^2$.

2. $(c-a{)}^2$.

3. $(a+2c{)}^2$.

4. $(3a-2b{)}^2$.

5. $(2a+3b{)}^2$.