1) To factor a polynomial by grouping, let's take 2 examples and do it step by step
2) Consider this quadratic equation, let's factor x² + 7x +12 by grouping
x² +7x +12 =0 Which two numbers whose sum is 7 and whose product is 12
S =___ + ___ = 7 4 + 3
P = ___x ___ = 12 4x 3
Then we can rewrite 7x as 4x +3x
x² +4x +3x +12 = 0 Let's group these terms into two parentheses:
(x² +4x) +(3x +12)
Now let's write them as factors, the GCD outside the parentheses x, 4x and 3, 12
x(x +4) +3(x +4)
Since we have a repetition of the factor (x +4) we can write it as a product
(x+4) (x+3) = x² +7x +12
Another example:
x³ -2x² +5x -10 Group them into two parentheses
(x³-2x²) +(5x -10) Factor out the GCD x³, 2x² =x² and 5, 10 =5
x²(x-2) +5(x-2) Now let's write them as a factor, (x -2x) is common to both
(x²+5) (x -2)
(x²+5) (x -2) = x³ -2x² +5x -10
3) So, in words.
1. Rewrite the polynomial into two parentheses grouping the terms
2. Factor out the GCD of the terms, leaving it outside the parentheses
3. Rewrite it as a product writing the outside factors into parentheses and the common term.