Case II.

107. When the terms can be grouped so as to show a common factor in each group.

1. Resolve into factors $ac+ad+bc+bd$.

The first two terms of $ac+ad+bc+bd$ are seen to have the common factor $a$, and the last two terms, the common factor $b$. Hence we bracket the first two terms and also the last two terms. Then we take out the factor $a$ from $(ac+ad)$ and $b$ from $(bc+bd)$, and get equation (2). Since one factor is seen in (2) to be $c+d$, dividing by $c+d$, we obtain the other factor, $a+b$.

2. Find the factors of $ac+ad-bc-bd$.

Here the last two terms, $-bc-bd$, being put within a parenthesis preceded by the sign $-$, have their signs changed to $+$.

3. Resolve into factors $2x^3-3x^2-4x+6$.

$$\begin{gathered} 2x^3-3x^2-4x+6\&=(2x^3-3x^2)-(4x-6) \\ \&=x^2(2x-3)-2(2x-3) \\ \&=(x^2-2)(2x-3). \end{gathered}$$

4. Resolve into factors $x^3+x^2-ax-a$.

$$\begin{gathered} x^3+x^2-ax-a\&=(x^3+x^2)-(ax+a) \\ \&=x^2(x+1)-a(x+1) \\ \&=(x^2-a)(x+1). \end{gathered}$$

5. Resolve into factors $x^3+3a{x}^2+x+3a$.

$$\begin{gathered} x^3+3a{x}^2+x+3a\&=(x^3+3a{x}^2)+(x+3a) \\ \&=x^2(x+3a)+1(x+3a) \\ \&=(x^2+1)(x+3a). \end{gathered}$$

Exercise 32.

Resolve into factors:

1. $x^3+x^2+x+1$.

2. $x^3-x^2+x-1$.

3. $x^2+xy+xz+yz$.

4. $ax-bx-ay+by$.

5. $a^2-ac+ab-bc$.

6. $x^2-bx+3x-3b$.

7. $2x^3-x^2+4x-2$.

8. $a^2-3a-ab+3b$.