Methods of Factoring
General Strategy for Factoring (Guess and Check)
Factor1) Arrange the polynomial in descending order. /
2) If the leading coefficient is negative, or a greatest common factor exists then factor this out from the entire polynomial (but don’t lose it!) /
3) To factor a trinomial of the form , you want to write it as a product of two binomials: Try combinations of the factors of a (in this case 3) in the first term of each binomial and combinations of c (in this case -7) in the second term of each binomial. If c is positive, the signs within the binomial factors match the sign of b; if c is negative, the signs within the binomial factors are opposite. /
So the answer is
4) Check by multiplying. / FOIL to get
Distribute the -1 to get
Factoring by Grouping
Factor1) Arrange the polynomial in descending order. /
2) If the leading coefficient is negative, or a greatest common factor exists then factor this out from the entire polynomial (but don’t lose it!) /
3) To factor a trinomial of the form , find two numbers (p and q) whose product is a*c (in this case -22) and whose sum is b (in this case -21). (if c is positive, the numbers have the same sign; if c is negative, the number have different signs) / p * q = -22
p + q = -21
p = -22
q = 1
5) Then rewrite the trinomial breaking the middle term into px + qx.
. / =
6) Then look at the first two terms [ ] and factor out of them both their greatest common factor. /
7) Repeat with the second two terms. If the trinomial is factorable then the binomials in brackets should be the same [in this case (x – 22)] /
8) Then factor this binomial out to get your final answer. /
9) Check by multiplying. / FOIL to get
Distribute the -2 to get
Table Method for Factoring
* Put the first term with its sign diagonally under the Factor and the third term with its sign diagonally under that, in the lower right hand corner. / Step 1
Factor
3a2
– 10 b2
* Get the diagonal product by multiplying the terms that are now in the table. / Step 2
Diagonal Product = (3a2)( – 10 b2) = –30a2 b2
* Write factors of the coefficient of the diagonal product. (Note: There are two sets because the product is negative.)
* Find the factors that add to the middle term (in this case +1) / Step 3
–1•30 / 1•(–30)
–2•15 / 2•(–15)
–3•10 / 3•(–10)
–5•6 / 5•(–6)
* Write these factors, with the variable(s) of the middle term of the original problem, in the lower left and upper right cells. / Step 4
Factor
3a2 / -5ab
6ab / – 10 b2
* Put the greatest common factor (gcf) of each row in the cell to the left of the row. / Step 5
Factor
gcf of 3a2 and -5ab is a® / a / 3a2 / -5ab
gcf of 6ab and – 10 b2is
2b ® / 2b / 6ab / – 10 b2
* Put the greatest common factor (gcf) of each column in the cell above the column.
* You can now read the factors from the outside of the table:
(a +2b) is in the left column
(3a-5b) is in the top row / gcf of 3a2 and 6ab is 3a
↓ / gcf of -5ab and – 10 b2 is -5b
↓
Factor / 3a / -5b
gcf of 3a2 and -5ab is a® / a / 3a2 / -5ab
gcf of 6ab and – 10 b2is
2b ® / 2b / 6ab / – 10 b2
Although this method is practically foolproof, you should always, always check by multiplying. / Step 6
Check:
(a +2b) (3a-5b) = 3a2 + ab – 10 b2