Math 10034 ebook Sec 1.2 Factor by Grouping
Factoring a Four – Term Polynomial by Grouping
Step 1. Arrange the terms so that the first two terms have a common factor and the last two terms have a
common factor.
Step 2. For each pair of terms, use the distributive property to factor out the pair’s greatest common
factor.
Step 3. If there is now a common binomial factor, factor it out.
Step 4. If there are no common binomial factor in step 3, begin again, rearranging the terms differently.
If no rearrangement leads to a common binomial factor, the polynomial cannot be factored.
Example 1) Factor 3x 2 + 4xy – 3x – 4y by grouping.
3x 2 + 4xy – 3x – 4y = x(3x + 4y) – (3x + 4y) = (3x + 4y)(x – 1)
Reminder: When factoring a polynomial, make sure the polynomial is written as a product. Do not write
the final answer as a sum or difference of terms like: x(3x+4y) - (3x+4y) This form is not a factored form
of the original polynomial. The factored form is the product: (3x + 4y)(x – 1)
Example 2) Factor ax – ab – 2bx + 2b 2 by grouping.
ax – ab – 2bx + 2b 2 = a(x – b) – 2b(x – b) = (x – b)(a -2b)
Sec 1.2 Exercises
Factor the four–term polynomial by grouping:
1. xy + 2x + 3y + 6
8. 3rs – s + 12r – 4
2. 2z – 8 + xz – 4x
9. 4b 2 – 2bc – 7cd + 14bd
3. a 3 + 4a 2 + a + 4
10. x 3 + 6x 2 – 4x – 24
4. 4b 2 – 8bc – 3b + 6c
11. a 3 – 2a 2 – 36a + 72
5. 3y – 5x + 15 – xy
12. x 3 – 28 + 7x 2 – 4x
6. 2x 3 – x 2 – 10x + 5
13. a 3 – 45 – 9a + 5a 2
7. 12a 2 b – 42a 2 – 4b + 14
14. -6 + 3x – 2x 2 + x 3
Answers: Sec 1.2 Factor by Grouping
1. (y + 2)(x + 3)
2. (z – 4)(2 + x)
3. (a + 4)(a 2 + 1)
4. (b – 2c)(4b – 3)
5. (3 – x)(y + 5)
6. (2x –1)(x 2 – 5)
7. 2(2b – 7)(3a 2 – 1)
8. (3r – 1)(s + 4)
9. (2b – c)(2b + 7d)
10. (x + 6)(x + 2)(x – 2)
11. (a – 2) (a + 6)(a – 6)
12. (x + 7)(x + 2)(x – 2)
13. (a + 5)(a +3)(a – 3)
14. (x – 2)(x 2 + 3)